Unlike stationary black holes the expanding universe with a cosmic curvature has a dynamically changing apparent horizon with the radius , where is a cosmic curvature [108] (see also [296]). Even in the FLRW spacetime, however, the Friedmann equation can be written in the thermodynamical form , where is the work density present in the dynamical background [8]. For matter contents of the universe with energy density and pressure , the work density is given by [297, 298]. This method is identical to the one established by Jacobson [324], that is, .

In metric f (R) gravity Eling et al. [228] showed that a non-equilibrium treatment is required such that the Clausius relation is modified to , where is the Wald horizon entropy [610] and is the bulk viscosity entropy production term. Note that corresponds to a Noether charge entropy. Motivated by this work, the connections between thermodynamics and modified gravity have been extensively discussed – including metric f (R) gravity [6, 7, 281, 431, 619, 620, 230, 103, 51, 50, 157] and scalar-tensor theory [281, 619, 620, 108].

Let us discuss the relation between thermodynamics and modified gravity for the following general action [53]

where is a kinetic term of a scalar field . For the matter Lagrangian we take into account perfect fluids (radiation and non-relativistic matter) with energy density and pressure . In the FLRW background with the metric , where and , with the two dimensional metric , the Friedmann equations are given by where and Note that and originate from the energy-momentum tensor defined by where the Einstein equation is given by Defining the density and the pressure of “dark” components in this way, they obey the following equation For the theories with (including f (R) gravity and scalar-tensor theory) the standard continuity equation does not hold because of the presence of the last term in Eq. (13.24).In the following we discuss the thermodynamical property of the theories given above. The apparent horizon is determined by the condition , which gives in the FLRW spacetime. Taking the differentiation of this relation with respect to and using Eq. (13.18), we obtain

In Einstein gravity the horizon entropy is given by the Bekenstein–Hawking entropy , where is the area of the apparent horizon [59, 75, 293]. In modified gravity theories one can introduce the Wald entropy associated with the Noether charge [610]:

Then, from Eqs. (13.25) and (13.26), it follows that The apparent horizon has the Hawking temperature , where is the surface gravity given by Here we have defined and . For the total equation of state less than 1/3, as is the case for standard cosmology, one has so that the horizon temperature is given by Multiplying the term for Eq. (13.27), we obtainIn Einstein gravity the Misner-Sharp energy [428] is defined by . In f (R) gravity and scalar-tensor theory this can be extended to [281]. Using this expression for theory, we have

where is the volume inside the apparent horizon. Using Eqs. (13.19) and (13.24), we find From Eqs. (13.30) and (13.32) it follows that Following [297, 298, 108] we introduce the work density . Then Eq. (13.33) reduces to which can be written in the form [53] where The modified first-law of thermodynamics (13.35) suggests a deep connection between the horizon thermodynamics and Friedmann equations in modified gravity. The term can be interpreted as a term of entropy production in the non-equilibrium thermodynamics [228]. The theories with lead to , which means that the first-law of equilibrium thermodynamics holds. The theories with , including f (R) gravity and scalar-tensor theory, give rise to the additional non-equilibrium term (13.36) [6, 7, 281, 619, 620, 108, 50, 53].The main reason why the non-equilibrium term appears is that the energy density and the pressure defined in Eqs. (13.20) and (13.21) do not satisfy the standard continuity equation for . On the other hand, if we define the effective energy-momentum tensor as Eq. (2.9) in Section 2, it satisfies the continuity equation (2.10). This correspond to rewriting the Einstein equation in the form (2.8) instead of (13.23). Using this property, [53] showed that equilibrium description of thermodynamics can be possible by introducing the Bekenstein–Hawking entropy . In this case the horizon entropy takes into account the contribution of both the Wald entropy in the non-equilibrium thermodynamics and the entropy production term.

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