  Recently, Savar\'{e}-Toscani proved that the R\'{e}nyi entropy power of
general probability densities solving the $p$-nonlinear heat equation in
$\mathbb{R}^n$ is always a concave function of time, which extends Costa's
concavity inequality for Shannon's entropy power to R\'{e}nyi entropies. In
this paper, we give a generalization of Savar\'{e}-Toscani's result by giving a
class of sufficient conditions of the parameters under which the concavity of
the R\'{e}nyi entropy power is still valid. These conditions are quite general
and include the parameter range given by Savar\'{e}-Toscani as special cases.
Also, the conditions are obtained with a systematical approach.
