Probing the thermodynamic properties of the Hawking-Page phase transition of
asymptotically-flat black holes in R\’enyi statistics. We reveal the evidence
for the persisting of the dual universal formula associated with the
Hawking-Page (HP) and minimum black hole thermodynamical transition points. Our
study unveils the universal properties of the black holes in different
statistics, in particular beyond the Gibbs-Boltzmann formalism, and motivates a
further promising bridge between the nonextensivity R\’enyi parameter {\lambda}
and the cosmological constant ? . Besides, we discover that universal ratios
related to horizon radius, R\’enyi temperature, and entropy can be shown to
appear in presence of such a transition.

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