Topology of the charged AdS black hole in restricted phase space

Abstract: The local topological properties of black hole systems can be expressed by the winding numbers as the defects. As so far, AdS black hole thermodynamics is often depicted by the dual parameters of $(T,S),~ (P,V), (\Phi, Q)$ in the extended phase space, while there is several study on the black hole thermodynamics in the restricted phase space. In this paper, we analyze the topological properties of the charged AdS black holes in the restricted phase space under the higher dimensions and higher order curvature gravities frame. The results show that the topological number of the charged black hole in the same canonical ensembles is a constant and is independent of the concrete dual thermodynamical parameters. However, the topological number in the grand canonical ensemble is different from that in the canonical ensemble for the same black hole system. Furthermore, these results are independent of the dimension $d$, the highest order $k$ of the Lanczos-Lovelock densities.