Abstract The time evolution of the Tsallis (T) and Renyi (R) entropies for a discrete state space was recently analyzed by Mariz [Phys. Lett. A 165 (1992) 409] and Ramshaw [Phys. Lett. A 175 (1993) 169] based on the master equation. Here we perform a corresponding analysis for a continuous state space χ in which the probability distribution ϱ(χ, t ) obeys the generalized Liouville equation. For this purpose it is necessary to formulate properly covariant generalizations of the T and R entropies in terms of ϱ(χ, t ). We show that if the microscopic dynamics is reversible in the Poincare-Lyapunov sense (i.e., D ( χ )=0, where D (χ) is the covariant divergence of the flow velocity in state space) then both the T and R entropies are constant in time, just like the conventional entropy. The T and R entropies are therefore not intrinsically irreversible. These results are obtained as special cases of a more general result: if D ( χ )=0 then d S d t =0 for any entropy functional S [ϱ(χ)] for which δS δϱ(χ) =ƒ( ϱ(χ) γ(χ) ) , where γ(χ) is the determinant of the metric tensor in state space and the function f is arbitrary.