Topological nature of black hole solutions in dRGT massive gravity

We study the thermodynamic properties of black holes in dRGT massive gravity theory using Duan's phi- mapping topological current theory. The topological features and the corresponding thermodynamic stability conditions for neutral and charged cases are discussed. A neutral black hole in four dimensions has a topological number 0, sharing the same topological class of D >= 6 Gauss-Bonnet-AdS black hole. We show that the charged black hole in four-dimensional massive gravity has the same topological structure as the AdS-RN black hole. Further, we have extended the calculations to higher dimensions. Our calculations strengthen the conjecture that the addition of higher interaction terms to Einstein-Hilbert action does not alter the topological number of black holes in four space-time dimensions. However, in higher-dimensional massive gravity, the topological number indeed depends on the black hole parameters.