Cosmology Due to Thermodynamics of Apparent Horizon

In this paper we propose new entropy of the apparent horizon $S_h=(1/\beta)\arctan(\beta S_{BH})$, where $S_{BH}$ is the Bekenstein--Hawking entropy. As parameter $\beta\rightarrow 0$ one comes to the Bekenstein--Hawking entropy. This allows us to consider the generalised Friedmann--Lema\^{i}tre--Robertson--Walker (FLRW) equations for the barotropic matter fluid with $p=w\rho$ for arbitrary equation of state parameter $w$. We obtain the matter pressure $p$ and density energy $\rho$ corresponding to the apparent horizon. The modified Friedmann's equations are found. The addition term in the second modified Friedmann's equation plays the role of a dynamical cosmological constant. The dark energy density, pressure and the deceleration parameter are found. It was shown that at some parameters $w$ and $\beta$ we can have two phases, acceleration and deceleration or the eternal inflation. The model under consideration by using the holographic principle describes the universe inflation. Thus, we consider the holographic dark energy model with the generalised entropy of the apparent horizon. We shown that entropic cosmology with our entropy proposed is equivalent to cosmology based on the teleparallel gravity with the function $F(T)$. New cosmology based on the generalized entropy can be of interest for a description of inflation and late time of the universe evolution.