Charged black hole chemistry with massive gravitons

In the subject of black hole chemistry, a broad variety of critical phenomena for charged topological black holes (TBHs) with massive gravitons (within the framework of dRGT massive gravity) is discussed in detail. Since critical behavior and nature of possible phase transition(s) crucially depend on the specific choice of ensemble, and, in order to gain more insight into criticality in the massive gravity framework, we perform our analysis in both the canonical (fixed charge, $Q$) and the grand canonical (fixed potential, $\Phi$) ensembles. It is shown that, for charged TBHs in the grand canonical ensemble, the van der Waals (vdW) phase transition could take place in $d \ge 5$, the reentrant phase transition (RPT) in $d \ge 6$ and the analogue of triple point in $d \ge 7$ which are different from the results of canonical ensemble. In the canonical ensemble, the vdW phase transition is observed in $d \ge 4$, the vdW type phase transition in $d \ge 6$ and the critical behavior associated with the triple point in $d \ge 6$. In this regard, the appearance of grand canonical $P-V$ criticality and the associated phase transition(s) in black holes with various topologies depend on the effective topological factor $k_{\rm{eff}}^{\rm{(GC)}} \equiv {[k + {m_g^2}c_0^2{c_2} - 2({d_3}/{d_2}){\Phi ^2}]}$ instead of $k$ in Einstein gravity, where $k$ is the normalized topological factor ($k_{\rm{eff}}^{\rm{(C)}} \equiv [k + {m_g^2}c_0^2{c_2}]$ plays this role in the canonical ensemble of TBHs in massive gravity). Such evidence gives the (grand) canonical study of extended phase space thermodynamics with massive gravitons a special significance.