In the present paper, we deal with asymptotical stability of Markov operators acting on abstract state spaces (i.e. an ordered Banach space, where the norm has an additivity property on the cone of positive elements). Basically, we are interested in the rate of convergence when a Markov operator T satisfies the uniform P-ergodicity, i.e. vertical bar vertical bar Tn - P vertical bar vertical bar -> 0, here P is a projection. We have showed that T is uniformly P-ergodic if and only if vertical bar vertical bar Tn - P vertical bar vertical bar = C beta(n), 0 < beta < 1. In this paper, we prove that such a beta is characterized by the spectral radius of T - P. Moreover, we give Deoblin's kind of conditions for the uniform P-ergodicity of Markov operators.