Thermodynamic Properties and Shadows of Black Holes in $f(R,T)$ Gravity

In this paper, we explore two $f(R,T)$ gravity models and derive black hole solutions within these models. We focus on investigating how the $f(R,T)$ model influences the thermodynamic characteristics of black holes by studying their thermodynamic topology and thermodynamic geometry. We consider five specific values of the thermodynamic parameter $\omega$, which signify five different classes of black hole solutions in general relativity (GR). We observe significant changes in the local topological properties of these black holes compared to GR, depending on the model parameters. Notably, we identify an additional topological class $W=0$ for some values of $\omega$ that is absent in the GR framework. We also study the thermodynamic geometry of the black hole using the Geometrothermodynamics (GTD) formalism. Our analysis demonstrates that the singular point, where the GTD scalar curvature diverges, corresponds exactly to the point where the heat capacity changes sign. Additionally, we constrain the model parameters of both models considered by utilizing black hole shadow data from the Sgr A* black hole, measured by the Event Horizon Telescope (EHT).