  In this work, we propose a common vector field to study the thermodynamic
topology of the Davies type and Hawking-Page phase transitions. Existing
literature has shown that studying these two types of phase transitions
typically requires defining two separate vector fields . In our approach, we
adopt Duan's $\phi$-mapping topological current theory to define a novel vector
field, denoted as $\phi$, whose critical points exactly correspond to the
Davies point and the Hawking-Page phase transition point. More importantly, we
can differentiate between these two points by their topological charge. While,
the topological charge for the critical point corresponding to the Davies-type
phase transition is found to be $-1$, the same for the Hawking-Page phase
transition point, it is $+1$. Although our analysis is applicable to all black
hole systems where both types of phase transitions are found, we illustrate it
using three simple systems as examples: the Schwarzschild AdS black hole, the
Reissner-Nordstr\"om AdS black hole in the grand canonical ensemble, and
finally the Kerr AdS black holes in the grand canonical ensemble. It is
wellknown that these black holes exhibit both Davies and Hawking-Page phase
transitions. With our proposed vector $\phi$, the critical points obtained for
these three systems exactly match the Davies-type and Hawking-Page phase
transition points, and the associated topological charges are found to be $-1$
for the Davies point and $+1$ for the Hawking-Page phase transition point.
