  Here, we consider a flat FRW universe whose its horizon entropy meets the
R\'enyi entropy of non-extensive systems. In our model, the ordinary
energy-momentum conservation law is not always valid. By applying the Clausius
relation as well as the Cai-Kim temperature to the apparent horizon of a flat
FRW universe, we obtain modified Friedmann equations. Fitting the model to the
observational data on current accelerated universe, some values for the model
parameters are also addressed. Our study shows that the current accelerating
phase of universe expansion may be described by a geometrical fluid, originated
from the non-extensive aspects of geometry, which models a varying dark energy
source interacting with matter field in the Rastall way. Moreover, our results
indicate that the probable non-extensive features of spacetime may also be used
to model a varying dark energy source which does not interact with matter
field, and is compatible with the current accelerated phase of universe.
