Cosmological evolution from modified Bekenstein entropy law

We study the dynamics of the homogeneous and isotropic cosmological background in the recently proposed ``quantum phenomenological gravitational dynamics'', characterised by logarithmic corrections to the Bekenstein entropy. We show that the model admits a family of solutions that are self-accelerating both at early and late times: they approach de Sitter in the future and admit a past attractor corresponding to an inflationary acceleration era. On the other hand, there are no solutions corresponding to a primordial bounce. We also show that asking scalar perturbations to be unaffected by instabilities on observable scales puts stringent constraints on the deviations from general relativity encoded by the model.