#### Abstract

This paper explores the nonequilibrium behavior of thermodynamics at the apparent horizon of isotropic and homogeneous universe model in gravity ( and represent the Gauss-Bonnet invariant and trace of the energy-momentum tensor, resp.). We construct the corresponding field equations and analyze the first as well as generalized second law of thermodynamics in this scenario. It is found that an auxiliary term corresponding to entropy production appears due to the nonequilibrium picture of thermodynamics in first law. The universal condition for the validity of generalized second law of thermodynamics is also obtained. Finally, we check the validity of generalized second law of thermodynamics for the reconstructed models (de Sitter and power-law solutions). We conclude that this law holds for suitable choices of free parameters.

#### 1. Introduction

The discovery of current cosmic accelerated expansion has stimulated many researchers to explore the cause of this tremendous change in cosmic history. A mysterious force known as dark energy (DE) is considered as the basic ingredient responsible for this expanding phase of the universe. Dark energy has repulsive nature with negatively large pressure but its complete characteristics are still unknown. Modified gravity theories are considered as the favorable and optimistic approaches to unveil the salient features of DE. These modified theories of gravity are obtained by replacing or adding curvature invariants as well as their corresponding generic functions in the Einstein-Hilbert action.

Gauss-Bonnet (GB) invariant is a linear combination of quadratic invariants of the formwhere , , and denote the Ricci scalar and Ricci and Riemann tensors, respectively. It is the second order Lovelock scalar with the interesting feature that it is free from spin-2 ghost instabilities while its dynamical effects do not appear in four dimensions [1–3]. The coupling of with scalar field and adding generic function in geometric part of the Einstein-Hilbert action are the two promising ways to study the dynamics of in four dimensions. Nojiri and Odintsov [4] presented the second approach as an alternative for DE referred as gravity which explores the fascinating characteristics of late-time cosmological evolution. This theory is consistent with solar system constraints and has a quite rich cosmological structure [5–7].

The curvature-matter coupling in modified theories has attained much attention to discuss the cosmic accelerated expansion. Harko et al. [8] introduced such coupling in gravity referred as gravity. Recently, we have established this coupling between quadratic curvature invariant and matter named as theory of gravity and found that such coupling leads to the nonconservation of energy-momentum tensor [9]. Furthermore, the nongeodesic lines of geometry are followed by massive test particles due to the presence of extra force while dust particles follow geodesic trajectories. The stability of Einstein universe is analyzed for both conserved and nonconserved in this theory [10]. Shamir and Ahmad [11] applied Noether symmetry approach to construct some cosmological viable models in the background of isotropic and homogeneous universe. We have reconstructed the cosmic evolutionary models corresponding to phantom/nonphantom epochs, de Sitter universe, and power-law solution and analyzed their stability [12].

The significant connection between gravitation and thermodynamics is established after the remarkable discovery of black hole (BH) thermodynamics with Hawking temperature as well as BH entropy [13–16]. Jacobson [17] obtained the Einstein field equations using fundamental relation known as Clausius relation (, , and represent the entropy, Unruh temperature, and energy flux observed by accelerated observer just inside the horizon, resp.) together with the proportionality of entropy and horizon area in the context of Rindler space-time. Cai and Kim [18] showed that Einstein field equations can be rewritten in the form of first law of thermodynamics for isotropic and homogeneous universe with any spatial curvature parameter. Akbar and Cai [19] found that the Friedmann equations at the apparent horizon can be written in the form (, , and are the energy, volume inside the horizon, and work density, resp.) in general relativity (GR), GB gravity, and the general Lovelock theory of gravity. In modified theories, an additional entropy production term appeared in Clausius relation that corresponds to the nonequilibrium behavior of thermodynamics while no extra term appears in GB gravity, Lovelock gravity, and braneworld gravity [20–24].

The generalized second law of thermodynamics (GSLT) has a significant importance in modified theories of gravity. Wu et al. [25] derived the universal condition for the validity of GSLT in modified theories of gravity. Bamba and Geng [26] found that GSLT in the effective phantom/nonphantom phase is satisfied in gravity. Sadjadi [27] studied the second law as well as GSLT in gravity for de Sitter universe model as well as power-law solution with the assumption that apparent horizon is in thermal equilibrium. Bamba and Geng [28] found that GSLT holds for the FRW universe with the same temperature inside and outside the apparent horizon in generalized teleparallel theory. Sharif and Zubair [29] checked the validity of first and second laws of thermodynamics at the apparent horizon for both equilibrium as well as nonequilibrium descriptions in gravity and found that GSLT holds in both phantom as well as nonphantom phases of the universe. Abdolmaleki and Najafi [30] explored the validity of GSLT for isotropic and homogeneous universe filled with radiation and matter surrounded by apparent horizon with Hawking temperature in gravity.

In this paper, we investigate the first as well as second law of thermodynamics at the apparent horizon of FRW model with any spatial curvature. The paper has the following format. In Section 2, we discuss the basic formalism of this gravity while the laws of thermodynamics are investigated in Section 3. Section 4 is devoted to analyze the validity of GSLT for reconstructed models corresponding to de Sitter and power-law solution. The results are summarized in the last section.

#### 2. Gravity

The action of gravity is given by [9]where , , and represent determinant of the metric tensor , gravitational constant, and matter Lagrangian density, respectively. The variation of the action (2) with respect to gives the fourth-order field equations aswhere , , ( is a covariant derivative), and has the following expression [31]:The variation of with respect to yieldswhere we have used the fact that depends only on .

The covariant derivative of (3) givesThe nonzero divergence shows that the conservation law does not hold in this gravity due to the curvature-matter coupling. The above equations show that matter Lagrangian density and a generic function have a significant importance to discuss the dynamics of curvature-matter coupled theories. The particular forms of arewhere the first choice is considered as correction to gravity since it does not involve the direct nonminimal curvature-matter coupling while the second form implies direct coupling. Unlike gravity [8], the choice ( is an arbitrary parameter) does not exist in this gravity since is a topological invariant in four dimensions. It is clear from (3) that the contribution of GB disappears for this particular choice of the model.

The energy-momentum tensor for perfect fluid as cosmic matter contents is given bywhere , , and denote pressure, energy density, and four-velocity, respectively. This four-velocity satisfies the relations and . In this case, the tensor with takes the formUsing (8) and (9), (3) can be written in a similar form as the Einstein field equations for dust case where

The line element for FRW universe model iswhere and and represent the scale factor depending on cosmic time and spatial curvature parameter which corresponds to open , closed , and flat geometries of the universe. The GB invariant takes the formUsing (8), (10), and (12), we obtain the following field equations:where is a Hubble parameter and dot represents derivative with respect to time. We can rewrite the above equations aswhere and are dark source terms given byThe continuity equation for (12) becomesThe conservation law holds in the absence of curvature-matter coupling for both gravity and GR.

#### 3. Laws of Thermodynamics

In this section, we study the laws of thermodynamics in the context of gravity at the apparent horizon of FRW universe model.

##### 3.1. First Law

The first law of thermodynamics is based on the concept that energy remains conserved in the system but can change from one form to another. To study this law, we first find the dynamical apparent horizon evaluated by the relationwhere is a two-dimensional metric. For isotropic and homogeneous universe model, the above relation gives the radius of apparent horizon asTaking the time derivative of this equation and using (16), it follows thatwhere represents the infinitesimal change in apparent horizon radius during the small time interval .

Bekenstein-Hawking entropy is defined as one-fourth of apparent horizon area in units of Newton’s gravitational constant [13–16]. In modified theories of gravity, the entropy of stationary BH solutions with bifurcate Killing horizons is a Noether charge entropy also known as Wald entropy [32]. It depends on the variation of Lagrangian density with respect to as [33–35]where and represent the volume element on -dimensional space-like bifurcation surface and binormal vector to satisfying the relation . Brustein et al. [36] proposed that Wald entropy is equal to quarter of in units of the effective gravitational coupling in modified theories of gravity. Using these concepts, the Wald entropy in gravity is given byThis formula corresponds to gravity for while the traditional entropy in GR is obtained for [37, 38]. Taking differential of (23) and using (21), we obtain

The surface gravity helps to determine temperature on the apparent horizon as [18]wherewhere is the determinant of . Using (24)–(26), we haveThe total energy inside the apparent horizon of radius for FRW universe model is given byThis equation shows that is directly related to , so the small displacement in horizon radius will cause the infinitesimal change given byUsing (27) and (29), it follows thatwhere is the work done by the system. The above equation can be written aswhereis interpreted as the entropy production term which appears due to nonequilibrium thermodynamical behavior at the apparent horizon. The nonequilibrium picture of thermodynamics implies that there is some energy change inside and outside the apparent horizon. Due to the presence of this extra term, the field equations do not obey the universal form of first law of thermodynamics in this gravity. It is mentioned here that, in modified theories, this auxiliary term usually appears in the first law of thermodynamics while it is absent in GR, GB gravity, and Lovelock gravity [19–24].

##### 3.2. Generalized Second Law

In this section, we discuss the GSLT in gravity which states that total entropy of the system is not decreasing in time given bywhere is the entropy due to energy as well as all matter contents inside the horizon and . The Gibbs equation relates to the total energy density and pressure as [25]where represents total temperature corresponding to all matter and energy contents inside the horizon and is not equal to the apparent horizon temperature. We assumeThis proportional relation shows that total temperature inside the horizon is positive and always smaller than the temperature at the apparent horizon. Using (31) and (34) in (33), we obtainwhereUsing (15) and (16), the GSLT condition takes the formwhereIt is seen that GSLT is valid for , and . For flat FRW universe model, the conditions , , , , and must be satisfied to protect the GSLT in gravity. The equilibrium description of thermodynamics implies that the temperature inside and at the horizon are the same yieldingThe validity of GSLT can be obtained for positive values of , , and .

#### 4. Validity of GSLT

Now we check the validity of GSLT for some reconstructed cosmological models in gravity.

##### 4.1. De Sitter Universe

The well-known cosmological de Sitter solution elegantly describes the evolution of current cosmic expansion. For this model, the Hubble parameter is constant and scale factor grows exponentially as . In case of dust fluid, energy density and GB invariant are given bywhere is an integration constant. In this case, (38) takes the formThe reconstructed model for de Sitter universe is given by [9]where ’s are integration constants and the standard conservation law is used in the reconstruction technique. The continuity constraint splits the above model into the following two forms:

Figures 1 and 2 show the validity of GSLT for the model (44) in the background of flat FRW universe model. The present day value of Hubble parameter is at the CL (CL stands for confidence level) which can be considered as in units of [39, 40]. The value of matter density parameter is constrained as with CL whereas scale factor at Gyr is [39]. For this model, we have four parameters , and with fixed values of , , and . Here, we examine the validity of GSLT against two parameters and with four possible choices of integration constants. For the case , we find that the validity of GSLT holds for the considered intervals of and . Figure 1(a) indicates the validity for while Figure 1(b) corresponds to the case and . Figure 2(a) shows that the validity of GSLT is not true for with while it satisfies for both negative values of as shown in Figure 2(b). It is found that the generalized second law holds at all times only for the same signatures of integration constants.

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The graphical behavior of GSLT for de Sitter model (45) is shown in Figures 3 and 4 against parameters and . In this case, we have considered , , and with four possible signature choices of integration constants and as in the previous model. Figures 3(b) and 4(a) show that GSLT is true for all considered values of and with opposite signatures of . For model (45), the choice of same signatures of integration constants is ruled out since it does not provide feasible region for which GSLT holds.

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##### 4.2. Power-Law Solution

Power-law solution has remarkable importance in modified theories of gravity to discuss the decelerated as well as accelerated cosmic evolutionary phases which are characterized by the scale factor as [41]The accelerated phase of the universe is observed for while covers the decelerated phase including dust as well as radiation dominated cosmic epochs. For this scale factor, the energy density and GB invariant becomesUsing (46) and (47) in (38), the validity condition for GSLT takes the formThe reconstructed model for dust fluid is given by [9]where ’s are integration constants andimply that the conservation law holds. The continuity constraint splits this model into two functions with some additional constants aswhere

The validity of GSLT for models (51) and (52) depends on five parameters , , , , and . Figures 5 and 6 show the behavior of GSLT for power-law reconstructed model (51). We examine the validity against and with , , , and while the behavior of remaining two integration constants and is investigated for four possible choices of their signatures. Figure 5 shows that GSLT is satisfied for both cases and with in the considered interval of parameters . We also check that the validity region decreases as the value of integration constants increases positively as well as negatively. Figure 6 shows that GSLT does not hold for model (51) when and with . From both figures, it is found that the signature of has dominant effect on the validity of GSLT as compared to .

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The validity of GSLT for the model (52) is shown in Figures 7 and 8. For this model, the viability of law again depends on five parameters , , , , and while we set , , , and . (a) shows that this law is satisfied for all values of at the initial times as well as when approaches to with for the case while the feasible region for and is shown in (b). Similarly, Figure 8 shows the regions where GSLT holds for the remaining two signatures of . In this case, we observe that validity of GSLT is true for all four possible choices of integration constants for the specific ranges of and .

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#### 5. Concluding Remarks

In this paper, we have investigated the first and second laws in the nonequilibrium description of thermodynamics and also checked the validity of GSLT for reconstructed models in gravity. The thermodynamical laws are studied at the apparent horizon of FRW universe model with any spatial curvature parameter . We have found that the total entropy in the first law of thermodynamics involves contribution from horizon entropy in terms of area and the entropy production term. This second term appears due to nonequilibrium behavior which implies that some energy is exchanged between outside and inside the apparent horizon. It is worth mentioning here that no such auxiliary entropy production term appears in GR, GB, Lovelock, and braneworld theories of gravity [19–24].

We have found the general expression for the validity of GSLT in terms of horizon entropy, entropy production term, and entropy corresponding to all matter and energy contents inside the horizon. For nonequilibrium picture of thermodynamics, it is assumed that temperature associated with all matter and energy contents inside the horizon is always positive and smaller than the temperature at apparent horizon. It is found that viability condition for this law is consistent with the universal condition for its validity in modified theories of gravity [25]. We have also investigated the validity condition of GSLT for the equilibrium description of thermodynamics. The validity of this law for the reconstructed models (de Sitter universe and power-law solution) for the dust fluid [9, 12] is also studied. The results can be summarized as follows.(i)For de Sitter reconstructed models, it is found that the validity of GSLT is true for model (44) when the integration constants have same signatures while, for the second model (45), the feasible regions are obtained for the opposite signatures (Figures 1–4).(ii)For power-law reconstructed models, the valid results are found when integration constant is positive for the model (51) while for, the model (52), this holds for all possible choices of and (Figures 5–8).

We conclude that the validity condition of GSLT is true for both reconstructed de Sitter and power-law models with suitable choices of free parameters.

#### Conflicts of Interest

The authors have no conflicts of interest.

#### Acknowledgments

The authors would like to thank the Higher Education Commission, Islamabad, Pakistan, for its financial support through the* Indigenous Ph.D. 5000 Fellowship Program Phase-II, Batch-III.*