  As we all know the local topological properties of thermodynamical systems
can be expressed by the winding numbers as the defects. The topological number
that is the sum of all winding numbers can be used to classify the global
topological nature of thermodynamical systems. In this paper, we construct a
kind of thermal potential and then put the Einstein-power-Yang-Mills AdS black
hole in it. Through the analysis of the geometric characteristics of the
thermal potential based on the complex analysis we find the topological number
is an invariant that is same as shown in the way of the Duan's $\phi$-mapping
topological current [Sci. Sin. 9, 1072 (1979)]. Furthermore, we adopt the
Kramer's escape rate method to investigate the intensity of the first-order
phase transition.
