  In this paper, a new family of Hawking-Page (HP) transition, the HP
transition with reentrance and triple point is introduced for the first time,
by investigating HP transition of hyperbolic AdS black hole in extended
thermodynamics of general dimensional Gauss-Bonnet gravity. The Reentrant HP
transition is composed of two HP transitions with a large and a small HP
temperature, and the triple point corresponds to small black holes/massless
black holes/large black holes (SBHs/MBHs/LBHs) phases all coexisting. We
present the two branches of HP transition temperatures, which both depend on
the pressure (i.e. the cosmological constant) and the Gauss-Bonnet constant.
Besides, the pressure and the Gauss-Bonnet constant both enlarge the large HP
temperature and diminish the small HP temperature. We also show the coexistence
lines in the $P-T$ phase diagrams. The triple point and an up critical point in
phase diagrams for arbitrary dimensional Gauss-Bonnet AdS black hole systems
are given, together with some interesting universal relations which only depend
on the dimensions of spacetime. We can explain the reentrant HP transition and
triple point of SBHs/MBHs/LBHs as the phase transition and triple point of
Hadronic matter/Quark-Gluon/Plasma/Quarkyonic matter in the QCD phase diagram.
These results may improve the comprehension of the black hole thermodynamics in
the quantum gravity framework and shed some light on the AdS/CFT correspondence
beyond the classical gravity limit.
