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In this work, we obtain the thermodynamic geometry of the extended black holesolution under f(R) gravity. This is the foundation of our work. We also obtainthe exact form of the f(R) model for some solutions. In the thermodynamicanalysis, we calculate the thermodynamic quantities such as temperature andentropy of these solutions and compare them with the corresponding BTZ blackholes. After that, we study the Ruppeiner geometry, such as the f(R)-gravity(extended black hole solution) of the BTZ black hole, and analyze thethermodynamic comparison of the Ruppeiner geometry with the van der Waalsequation for its phase transition process.

Thermodynamic geometric analysis of BTZ black hole in f(R,{\phi}) Gravity