Topological properties of black holes in five-dimensional gauged supergravity

In this paper, we study the topological properties of five-dimensional rotating charged black holes in different ensembles. The topological numbers for the black holes are gotten in the grand canonical and canonical ensembles, which are 1. When the charge and cosmological constant disappear, their topological numbers are 0. When the pressure is lower than the critical pressure, the phase transition exists in the canonical ensemble. However, the phase transition also exists in the grand canonical ensemble when the pressure is higher than the critical pressure and two independent rotational parameters are $a=1$ and $b=-1$. This may be due to the fact that the values of the rotational parameters change the supersymmetry of the black hole and lead to the phase transition.