  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.
