  We investigate the thermodynamic properties of the Hawking-Page phase
transition of AdS black holes. We present evidence for the existence of two
universal critical constants associated with the Hawking-Page (HP) and minimum
black hole thermodynamical transition points. These constants are defined by
C_S =\frac{S_{HP}-S_{min}}{S_{min}} and C_T =\frac{T_{HP}-T_{min}}{T_{min}}
where S_{min}(S_{HP}) and T_{min}(T_{HP}) are the minimal (HP phase transition)
entropy and temperature, respectively, below which no black hole can exist. For
a large class of four dimensional non-rotating black holes, we find C_S =2 and
C_T = \frac{2-\sqrt{3}}{\sqrt{3}}. For the rotating case, however, such
universal ratios are slightly affected without losing the expected values.
Taking small values of the involved rotating parameter, we recover the same
constants. Higher dimensional models, with other universal constants, are also
discussed in some details.
