  We investigate the thermodynamic curvature in the Hawking-Page phase
transition and the second-order phase transition of the AdS black hole. It is
shown that the thermodynamic curvature has the same behavior in these two
different phase transitions. Specifically, the thermodynamic curvature in the
Hawking-Page phase transition is the power function of the Hawking-Page phase
transition temperature with exponent related to the spacetime dimension, or is
invariably proportional to the reciprocal of the Hawking-Page phase transition
entropy. For the thermodynamic curvature in the second-order phase transition,
the conclusion is similar. The more enlightening result is that the ratio of
thermodynamic curvatures along the two different phase transitions tends to the
natural constant in the limit that the number of spacetime dimensions is
infinite. The universal and novel result provides an essential difference
between the Hawking-Page phase transition and the second-order phase transition
of the AdS black hole in the large dimension paradigm.
