Topological classes of black holes in de-Sitter spacetime

In this paper, we investigate the topological number of de-Sitter black hole solutions with different charges $(q)$ and rotational $(a)$ parameters. By using generalized free energy and Duan's $\phi$-mapping topological current theory, we find that the topological numbers of black holes can still be classified as three types. In addition, we interestingly found the topological classes for de-Sitter $($dS$)$ spacetime with distinct horizon, i.e, black hole event horizon and cosmological horizon, will be different. Moreover, we also investigate topological classifications of dS black hole solutions in higher dimensions with or without Gauss-Bonnet term.