  Universal relations and constants have important applications in
understanding a physical theory. In this article, we explore this issue for
Hawking-Page phase transitions in Schwarzschild anti-de Sitter black holes. We
find a novel exact dual relation between the minimum temperature of the
($d$+1)-dimensional black hole and the Hawking-Page phase transition
temperature in $d$ dimensions, reminiscent of the holographic principle.
Furthermore, we find that the normalized Ruppeiner scalar curvature is a
universal constant at the Hawking-Page transition point. Since the Ruppeiner
curvature can be treated as an indicator of the intensity of the interactions
amongst black hole microstructures, we conjecture that this universal constant
denotes an interaction threshold, beyond which a virtual black hole becomes a
real one. This new dual relation and universal constant are fundamental in
understanding Hawking-Page phase transitions, and might have new important
applications in the black hole physics in the near future.
