Thermodynamic behavior for generalized f(R) gravity with arbitrary coupling between matter and geometry

The thermodynamic behavior of field equations for the generalized f(R) gravity with arbitrary coupling between matter and geometry is studied in the two kinds of spacetime, i.e., the both spatially homogenous, isotropic FRW universe and static, spherically symmetric black hole spacetime. The field equations of the generalized f(R) gravity with arbitrary coupling between matter and geometry can be cast to the form of the first law of thermodynamics with the reputed entropy production terms d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\bar S$ \end{document}, which are quite general and can degenerate to the cases of Einstein’s general relativity and pure f(R) gravity with non-coupling and nonminimal coupling as special cases. The appearance of the entropy production term d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\bar S$ \end{document} illustrates that the horizon thermodynamics is non-equilibrium one for the generalized f(R) gravity with arbitrary coupling between matter and geometry.