Thermodynamic geometric analysis of RN black holes under <i>f</i> ( <i>R</i> ) gravity

This paper investigates the thermodynamic phase structure and microscopic interactions of Reissner–Nordström (RN) black holes within the framework of f( R) gravity. The academic rationale is to explore how modifications to Einstein’s gravity influence the critical behavior of charged black holes. We employ the tools of thermodynamic geometry, specifically the Ruppeiner curvature, to probe the nature of interactions among the black hole’s microscopic constituents. Our analysis reveals that the thermodynamic behavior is highly sensitive to the functional form of f( R) and the background space–time geometry. For models with a constant initial curvature scalar, RN black holes exhibit thermodynamic properties analogous to an ideal gas. However, for models where the initial curvature scalar is non-constant, corresponding to a cosmological constant with negative power terms, we uncover a first-order phase transition analogous to that of a Van der Waals fluid. This transition is characterized by the canonical “swallowtail” catastrophe in the Gibbs free energy landscape. The divergence of the Ruppeiner curvature scalar is shown to precisely coincide with the critical points of this phase transition, providing a geometric signature of the underlying change in the black hole’s microstructure. Our results demonstrate that f( R) gravity parameters and background topology are critical determinants of black hole stability, offering a new window into the phenomenological consequences of modified gravity theories.