Probing Black Hole Thermodynamics and Microstructure via the Shadow of Sagittarius A*

We explore the connection between black hole shadows, thermodynamic phase structure, and microstructure of charged and rotating black holes within General Relativity and Geometrothermodynamics. Focusing on Reissner-Nordstr\"om and Kerr solutions, we establish a criterion to select the most suitable Geometrothermodynamic metric for a system, revealing that the first metric from enthalpy and the second from mass correctly reproduce heat capacity singularities. We show that the shadow radius encodes the same phase information as entropy and introduce Shadow-Microstructure diagrams to extract insights into stability and microscopic interaction types directly from observational bounds. Applying this framework to Sagittarius A*, we constrain the macroscopic parameters and the allowed microscopic thermodynamic phases. Our findings indicate that shadow measurements offer a novel probe of thermodynamic and microscopic aspects of black holes, enabling tests of alternative theories of gravity and thermodynamic frameworks.