Topology of black hole thermodynamics in Lovelock gravity

In this work, we present a convenient method to perform the topological analysis of black hole thermodynamics. Utilizing the spinodal curve, thermodynamic critical points of a black hole are endowed with a topological quantity, Brouwer degree, which reflects intrinsic properties of the system under smooth deformations. Specially, in our setup, it can be easily calculated without exact solution of critical points. This enables us to conveniently investigate the topological transition between different thermodynamic systems, and give a topological classification for them. In this framework, topology of Lovelock AdS black holes with spherical horizon geometry is explored. Results show that charged black holes in arbitrary dimensions can be classified into the same topology class, whereas the $d=7$ and $d \geq 8$ uncharged black holes are in different topology classes. Moreover, we revisit the relation between different phase structures of these black holes from the viewpoint of topology. Some general topological properties of critical points are also discussed.