Photon Spheres and shadow of modified black-hole entropies

Starting from the first law of black hole thermodynamics, we establish an explicit correspondence between the corrected entropy and the metric function under the condition of fixed black hole energy and horizon position. Using the corrected metric, we further compute the photon sphere radius and shadow size, demonstrating that different entropy corrections lead to characteristic optical shifts. By comparing with the Event Horizon Telescope observations of Sgr A*, we constrain the parameter range introduced in the corrected entropy. This provides a feasible approach for testing generalized entropy frameworks and probing deviations from the Bekenstein-Hawking area law.