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Massive gravities in anti-de Sitter spacetime can be viewed as effective dualfield theories of different phases of condensed matter systems with brokentranslational symmetry such as solids, (perfect) fluids, and liquid crystals.Motivated by this fact, we explore the black hole chemistry (BHC) of thesetheories and find a new range of novel phase transitions close to realisticones in ordinary physical systems. We find that the equation of state oftopological black holes (TBHs) at their inflection point(s) in $d$-dimensionalspacetime reduces to a polynomial equation of degree $(d-4)$, which yields upto $n=(d-4)$ critical points. As a result, for (neutral) TBHs, we observetriple-point phenomena with the associated first-order phase transitions (in $d\ge 7$), and a new phenomenon we call an $N$-fold reentrant phase transition,in which several ($N$) regions of thermodynamic phase space exhibit distinctreentrant phase transitions, with associated virtual triple points andzeroth-order phase transitions (in $d \ge 8$), as well as Van der Waalstransitions (in $d \ge 5$) and reentrant (in $d \ge 6$) behavior. We concludethat BHC in higher-dimensional massive gravity is very likely to offer furthernew surprises.

Range of novel black hole phase transitions via massive gravity: Triple points and N-fold reentrant phase transitions