Thermodynamics in f(R,RαβRαβ,ϕ) theory of gravity

First and second laws of black hole thermodynamics are examined at the apparent horizon of FRW spacetime in f(R,RαβRαβ,ϕ) gravity, where R, RαβRαβ and ϕ are the Ricci invariant, Ricci tensor and the scalar field respectively. In this modified theory, Friedmann equations are formulated for any spatial curvature. These equations can be presented into the form of first law of thermodynamics for ThdSˆh+ThdiSˆh+WdV=dE, where diSˆh is an extra entropy term because of the non-equilibrium presentation of the equations and ThdSˆh+WdV=dE for the equilibrium presentation. The generalized second law of thermodynamics (GSLT) is expressed in an inclusive form where these results can be represented in GR, f(R) and f(R,ϕ) gravities. Finally to check the validity of GSLT, we take some particular models and produce constraints of the parameters.