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First and second laws of black hole thermodynamics are examined at theapparent horizon of FRW spacetime in $f(R,R_{\alpha\beta}R^{\alpha\beta},\phi)$ gravity, where $R$, $R_{\alpha\beta}R^{\alpha\beta}$ and $\phi$ are theRicci scalar, Ricci invariant and the scalar field respectively. In thismodified theory, Friedmann equations are formulated for any spatial curvature.These equations can be presented into the form of first law of thermodynamicsfor $T_{h}d\hat{S}_{h}+ T_{h}d_{i}\hat{S}_{h}+W dV=dE$, where$d_{i}\hat{S}_{h}$ is an extra entropy term because of the non-equilibriumpresentation of the equations and $T_{h}d\hat{S}_{h}+W dV=dE$ for theequilibrium 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 $f(R, \phi)$ gravities. Finally to check the validity of GSLT, wetake some particular models and produce constraints of the parameters.

Thermodynamics in $f(R,R_{\alpha\beta}R^{\alpha\beta},\phi)$ theory of gravity