Abstract The Rényi statistics in the canonical and microcanonical ensembles is examined both in general and in particular for the ideal gas. In the microcanonical ensemble the Rényi statistics is equivalent to the Boltzmann–Gibbs statistics. By the exact analytical results for the ideal gas, it is shown that in the canonical ensemble, taking the thermodynamic limit, the Rényi statistics is also equivalent to the Boltzmann–Gibbs statistics. Furthermore it satisfies the requirements of the equilibrium thermodynamics, i.e. the thermodynamical potential of the statistical ensemble is a homogeneous function of first degree of its extensive variables of state. We conclude that the Rényi statistics arrives at the same thermodynamical relations, as those stemming from the Boltzmann–Gibbs statistics in this limit.