The description of the accelerated phases for the current and primordial cosmos has been obtained in the framework of R´enyi cosmology. The results show that corrections to the Friedmann equations, originated from differences between R´enyi and Bekenstein entropies, can describe both the current accelerated and inflationary eras. The effects of these corrections can be appointed to a hypothetical fluid which is classically unstable. Although inflation can be obtained in R´enyi cosmology as the vacuum solution, the values found for the parameters of the hypothetical fluid which supports the current accelerated and primary inflationary cosmos differ from each other.
