  We use the causal horizon entropy to study the asymptotic behaviors of
regular AdS black holes. In some literature, the causal horizon entropy is
regarded as a generalized holographic $c$-function. In this paper, we apply
this idea to the case of regular AdS black holes. We show that the causal
horizon entropy decreases to zero at the center of regular AdS black holes and
in particular it is stationary because its derivative with respect to the
affine parameter approaches zero asymptotically. Meanwhile, the asymptotic
behavior of the metric of regular AdS black holes implies that the black hole
center corresponds to an IR fixed point. Therefore, we conclude that the causal
horizon entropy is a valid candidate for the holographic $c$-function of these
regular AdS black holes.
