  Recently, a novel emergent phase can occur from thermodynamic consideration
of the asymptotically flat Reissner-Nordstr\"om black hole (RN-AF) using
R\'enyi statistics. We present an analysis of the thermodynamical and
mechanical stabilities of the RN-AF in both the Gibbs-Boltzmann (GB) and the
alternative R\'enyi statistics when charge $q$ and electrostatic potential
$\phi$ are treated as pressure and volume, respectively. Interestingly, the
emergent phase of the RN-AF can be both thermodynamically and mechanically
stable in some range of parameters in the framework of R\'enyi thermodynamics.
With the construction of the Maxwell equal area law in $q-\phi$ plane, the
coexistence line between the near-extremal black hole phase and the emergent
phase can be found in some values of charge which can be associated as the
vapor pressure at which the liquid and gas phases coexist. In the aspect of
thermodynamic geometry, the microscopic interaction between the black hole
microstructures can be repulsive in the R\'enyi description. This implies that
a novel correlation between the microstates of a self-gravitating system could
be emerged via the nonextensive nature of long-range interaction systems.
Finally, we also investigate the critical phenomena of the RN-AF in R\'enyi
statistics compared to that of the van der Waals (vdW) fluid and find that the
critical exponents of the relevant physical quantities of both systems are
identical. This implies that both systems are in the same universality class of
the phase transition.
