  Considering that under the framework of the massive gravity theory, the
interaction between the mass gravitons and Schwarzschild black hole (BH) could
make it carry a scalar charge, the phase transition process caused by this
scalar charge is investigated in this analysis. The phase transition grade and
microstructure of those BHs are investigated from both macroscopic and
microscopic points of view. From the macroscopic point of view, it is found
that Ehrenfest equations are satisfied at the phase transition critical point,
which implies that the phase transition grade of those BHs is second-order.
Based on the BH molecules model and Landau continuous phase transition theory,
the phase transition of those BHs from the microcosmic point of view is
analyzed. The critical exponents obtained from the two perspectives are
consistent. By investigating the Ruppeiner geometry, the microstructure feature
of those BHs is revealed. These results suggest that the phase transition of BH
in massive gravity is a standard second-order phase transition at the critical
point, and the microscopic details of those BHs are different from the RN-AdS
BH in standard Einstein gravity.
