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

General Relativity and Quantum Cosmology