New entropy à   la Born–Infeld, thermodynamics of apparent horizon and cosmology

New one-parameter entropy with the function [Formula: see text] with [Formula: see text] being the Bekenstein–Hawking entropy is proposed. At the limit [Formula: see text], [Formula: see text] becomes the Bekenstein–Hawking entropy. The spatial flat Friedmann–Lemaître–Robertson–Walker universe is considered. We study the entropic cosmology with the barotropic matter fluid. The modified Friedmann equations and the effective (dynamical) cosmological constant are obtained. From Friedmann’s equations the dark energy density, pressure and the deceleration parameter are obtained. It is demonstrated that at some parameters [Formula: see text] and [Formula: see text] one has two phases, acceleration and deceleration or the eternal inflation. At the entropy parameter [Formula: see text], where [Formula: see text] is the current Hubble rate, the normalized density of the matter field at the current time [Formula: see text] and the deceleration parameter [Formula: see text] take place according to the Planck data. The model under consideration using the entropic cosmology can describe the universe inflation and the current universe acceleration. We show that the entropic cosmology under consideration corresponds to the certain teleparallel gravity with the torsion. The Hubble parameters within [Formula: see text]% are in approximate agreement with the observational Hubble data for redshifts [Formula: see text].