BTZ Black Hole in the Nonextensive Generalizations of Gibbs Entropy

We study the thermodynamics and thermodynamic geometry of the (2+1)D Banados-Teitelboim-Zanelli (BTZ) black hole within the framework of the nonextensive generalizations of Gibbs entropy. We investigate both the rotating (R-BTZ) and the charged (C-BTZ) BTZ black holes in these nonextensive entropy formalisms. We write down the Bekenstein-Hawking (BH) entropy of the black hole in terms of the nonextensive entropies, namely Kaniadakis entropy, Renyi entropy, and Barrow entropy. We investigate their impact on the thermodynamic phase structure and geometry of the BTZ black holes in both the ensembles, i.e. the fixed $(J)$ and fixed $(\Omega )$ ensembles for the R-BTZ black hole and the fixed $(Q)$ and fixed $(\Phi )$ ensembles for the C-BTZ black hole, where $J$, $\Omega$, $Q$, and $\Phi$ represent the angular momentum, angular velocity, charge, and the electric potential of the respective black holes. We investigate the Ruppeiner and geometrothermodynamic (GTD) geometries of the black hole for all the nonextensive entropy cases. We find that there are both Davies-type and Hawking-Page phase transitions in both the charged and rotating BTZ black holes for the Kaniadakis entropy case in all the above-mentioned thermodynamic ensembles. These phase transitions were not seen in the BH entropy case. We also find that the Ruppeiner and the GTD scalar for the Kaniadakis entropy show curvature singularities corresponding to Davies-type phase transitions in both the rotating and charged BTZ black holes.