  The coexistent physics of de Rham-Gabada-dze-Tolley (dRGT) massive black
holes and holographic massive black holes is investigated in the extended phase
space where the cosmological constant is viewed as pressure. Van der Waals like
phase transitions are found for both of them. Coexistent curves of reduced
pressure and reduced temperature are found to be different from that of RN-AdS
black holes. Coexistent curves of reduced Gibbs free energy and reduced
pressure show that Gibbs free energy in the canonical ensemble decreases
monotonically with the increasing pressure. The concept number density is
introduced to study the coexistent physics. It is uncovered that with the
increasing pressure, the number densities of small black holes (SBHs) and large
black holes (LBHs) change monotonically in the contrary directions till finally
reaching the same value at the critical points of the phase transitions. In
other words, with the increasing pressure the number density differences
between SBHs and LBHs decrease monotonically before disappearance at the
critical points. Further more, HEPM thermodynamic scalar curvature differences
between SBHs and LBHs are found to decrease monotonically to zero when
approaching to the critical points, which is similar as a RN-AdS black hole. We
propose that both the number density difference and the HEPM scalar curvature
difference can be order parameters describing the SBH/LBH phase transition and
judging the upcoming of critical point where a second-order phase transition
takes place. These results provide us with new recognition of the massive
gravity. The thermodynamics in the extended phase space of AdS black holes is
enriched.
