  Topological arguments are currently being used as a novel scheme to discern
the properties of black holes while ignoring their detailed structure and
specific field equations. Among various avenues of black hole physics, where
this novel approach is being utilized, the phase transition in black hole
thermodynamics lies at the forefront. There are several types of phase
transition in black holes; such as the van der Waals type phase transition,
Davies-type phase transition, extremal phase transition, and Hawking-Page (HP)
transition. So far, the topological interpretation, where the critical point
has been identified with the non-zero topological charge, has been obtained
only for the van der Waals type phase transition and HP transition in different
spacetimes. To complete the picture, here we provide the same interpretation
for two other phase transitions: Davies-type phase transition and extremal
phase transition. The entire analysis is general and is valid for any spacetime
where these types of phase transitions are observed. More importantly, our
analysis suggests that amid the apparent differences in these phase
transitions, they share the same topological characteristics, \textit{i.e.}
non-zero topological charge arising from different thermodynamic potentials in
different types of phase transition.
