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

Topology and phase transition for EPYM AdS black hole in thermal potential