  In this note, we obtain the asymptotic estimate for the time derivative of
the $\Phi$-entropy in terms of the lower bound on the Bakry-Emery $\Gamma_2$
curvature. In the cases of Hyperbolic space and Heisenberg group, we show that
the time derivative of the $\Phi$-entropy is non-increasing, and we also get
sharp asymptotic bound for the time derivative of the entropy in the Heisenberg
group.
