  In this work, employing the generalized off-shell free energy, we treat black
hole solutions as defects in the thermodynamic parameter space. The results
show that the positive and negative winding numbers corresponding to the
defects indicate the local thermodynamical stable and unstable black hole
solutions, respectively. The topological number $W$ defined as the sum of the
winding numbers for all the black hole branches at an arbitrary given
temperature is found to be a universal number independent of the black hole
parameters. Moreover, this topological number only depends on the thermodynamic
asymptotic behaviors of the black hole temperature at small and large black
hole limits. Different black hole systems are characterized by three classes
via this topological number. This number could help us well understanding the
black hole thermodynamics, and further shed new light on the fundamental nature
of quantum gravity.
