Thermodynamics of apparent horizon and modified Friedmann equations

Starting from the first law of thermodynamics, dE=Th  dSh+W  dV, at the apparent horizon of a FRW universe, and assuming that the associated entropy with apparent horizon has a quantum-corrected relation, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S=\frac{A}{4G}-\alpha \ln \frac{A}{4G}+\beta \frac{4G}{A}$\end{document}, we derive modified Friedmann equations describing the dynamics of the universe with any spatial curvature. We also examine the time evolution of the total entropy including the quantum-corrected entropy associated with the apparent horizon together with the matter field entropy inside the apparent horizon. Our study shows that, with the local equilibrium assumption, the generalized second law of thermodynamics is fulfilled in a region enclosed by the apparent horizon.