statistical mechanics - Motivation for maximum Renyi/Tsallis entropy

The Conditional limit theorem of Van Campenhout and Cover gives a physical reason for maximizing (Shannon) entropy. Nowadays, in statistical mechanics, people talk about maximum Renyi/Tsallis entropy distributions. Is it just because these distributions are heavy tailed?

Is there any motivation (or physical significance) for maximizing Renyi/Tsallis entropies?

Answer

Warning: I do not work in statistical physics, and I therefore do not now if the following has any link with the reason people use Rényi entropies in statistical physics.

John C. Baez wrote a paper (arXiv:1102.2098), stating that the Rényi entropy $H_\beta$ is proportinal to the free energy at temperature defined by $\beta=T_0/T$. This paper is nicely explained on his blog.

The free energy of a system kept at a constant volume and temperature is minimal, and this corresponds, when $T>T_0$ to a maximum Rényi entropy $H_{T/T_0}$.

Edited after reading the paper to correct some mistakes.