  Based on the relationship between thermodynamics and gravity we propose, with
the aid of Verlinde's formalism, an alternative interpretation of the dynamical
evolution of the Friedmann-Robertson-Walker Universe. This description takes
into account the entropy and temperature intrinsic to the horizon of the
universe due to the information holographically stored there through
non-gaussian statistical theories proposed by Tsallis and Kaniadakis. The
effect of these non-gaussian statistics in the cosmological context is change
the strength of the gravitational constant. In this paper, we consider the
$w$CDM model modified by the non-gaussian statistics and investigate the
compatibility of these non-gaussian modification with the cosmological
observations. In order to analyze in which extend the cosmological data
constrain these non-extensive statistics, we use type Ia supernovae, baryon
acoustic oscillations, Hubble expansion rate function and the linear growth of
matter density perturbations data.
