Thermodynamic laws for generalized f(R) gravity with curvature-matter coupling

The first law and the generalized second law (GSL) of thermodynamics for the generalized f(R) gravity with curvature-matter coupling are studied in the spatially homogeneous, isotropic FRW universe. The research results show that the field equations of the generalized f(R) gravity with curvature-matter coupling can be cast to the form of the first law of thermodynamics with the so-called the entropy production terms dS¯ and the GSL can be given by considering the FRW universe filled only with ordinary matter enclosed by the dynamical apparent horizon with the Hawking temperature. Furthermore, as a concrete example, by utilizing the GSL the constraints on the gravitational model with f1(R)=R+αRl and f2(R)=Rm are also discussed. It is worth noting these results given by us are quite general and can degenerate to the ones in Einsteinʼs general relativity and pure f(R) gravity with non-coupling and non-minimal coupling as special cases. Comparing with the case of Einsteinʼs general relativity, the appearance of the entropy production term dS¯ in the first law of thermodynamics demonstrates that the horizon thermodynamics is non-equilibrium one for generalized f(R) gravity with curvature-matter coupling, which is consistent with the arguments given in Akbar and Cai (2007) [13] and Eling et al. (2006) [18].