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 topological current $\phi-$ mapping theory. The topological features and the corresponding thermodynamic stability conditions for neutral and charged cases are discussed. The uncharged black hole in four dimensions has a topological number $0$, sharing the same topological class of $D\geq6$ 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 spacetime dimensions. However, in higher dimensional massive gravity, the topological number indeed depends on the black hole parameters.