Apparent horizon and gravitational thermodynamics of the Universe:   Solutions to the temperature and entropy confusions, and extensions to   modified gravity
  The thermodynamics of the Universe is restudied by requiring its compatibility with the holographic-style gravitational equations which govern the dynamics of both the cosmological apparent horizon and the entire Universe, and possible solutions are proposed to the existent confusions regarding the apparent-horizon temperature and the cosmic entropy evolution. We start from the generic Lambda Cold Dark Matter ($\Lambda$CDM) cosmology of general relativity (GR) to establish a framework for the gravitational thermodynamics. The Cai-Kim Clausius equation for the isochoric process of an instantaneous apparent horizon indicates that, the Universe and its horizon entropies encode the \emph{positive heat out} thermodynamic sign convention, which encourages us to adjust the traditional positive-heat-in Gibbs equation into the positive-heat-out version $dE_m=-T_mdS_m-P_mdV$. It turns out that the standard and the generalized second laws (GSLs) of nondecreasing entropies are always respected by the event-horizon system as long as the expanding Universe is dominated by nonexotic matter $-1\leq w_m\leq 1$, while for the apparent-horizon simple open system the two second laws hold if $-1\leq w_m<-1/3$; also, the artificial local equilibrium assumption is abandoned in the GSL. All constraints regarding entropy evolution are expressed by the equation of state parameter, which show that from a thermodynamic perspective the phantom dark energy is less favored than the cosmological constant and the quintessence. Finally, the whole framework is extended from GR and $\Lambda$CDM to modified gravities with field equations $R_{\mu\nu}-Rg_{\mu\nu}/2=8\pi G_{\text{eff}} T_{\mu\nu}^{\text{(eff)}}$. Furthermore, this paper argues that the Cai-Kim temperature is more suitable than Hayward, and the Bekenstein-Hawking and Wald entropies cannot unconditionally apply to the event and particle horizons. 
David Wenjie Tian, Ivan Booth
gr-qc astro-ph.CO