Recently, Savaré-Toscani proved that the Rényi 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 Rényi entropies. In this paper, we give a generalization of Savaré-Toscani's result by giving a class of sufficient conditions of the parameters under which the concavity of the Rényi entropy power is still valid. These conditions are quite general and include the parameter range given by Savaré-Toscani as special cases. Also, the conditions are obtained with a systematical approach.