The R\'{e}nyi entropy and entropic cosmology

Entropic cosmology with the R\'{e}nyi entropy of the apparent horizon $S_R=(1/\alpha)\ln(1+\alpha S_{BH})$, where $S_{BH}$ is the Bekenstein--Hawking entropy, is studied. By virtue of the thermodynamics-gravity correspondence a model of dark energy is investigated. The generalised Friedmann equations for the Friedmann--Lema\^{i}tre--Robertson--Walker spatially flat universe with the barotropic matter fluid are obtained. We compute the dark energy density $\rho_D$, pressure $p_D$ and the deceleration parameter $q$ of the universe. At some model parameters the normalized density parameter of the matter $\Omega_{m0}\approx 0.315$ and the deceleration parameter $q_0\approx -0.535$ for the current epoch, which are in the agreement with the Planck data, are found. Making use of the thermodynamics-gravity correspondence, we describe the late time of the universe acceleration. The entropic cosmology considered is equivalent to cosmology based on the teleparallel gravity with the definite function $F(T)$. The Hubble parameters are in approximate agreement (within $5$ percents) with the observational Hubble data for redshifts $0.07\leq z \leq 1.75$ at the entropy parameter $\alpha\approx 0.305~GH_0^2$.