Analyzing thermodynamic observables and geometries through modified entropy

In this paper, we explore the thermodynamic properties of the Schwarzschild AdS 5 × S 5 black hole with the inclusion of three entropy parameters. The thermodynamic properties of temperature, Gibbs free energy, chemical potential, heat capacities, geometry and sparsity index were examined in detail in terms of the modified entropy corrections parameter β . It is seen that the parameter β plays a significant role in the thermodynamic properties compared to the other corrected parameters α and γ . The temperature decreases with entropy, with larger values of β reducing black hole radiation. The Gibbs free energy shows typical patterns that define phase transitions, together with negative chemical potential, which confirms complete system stability. The two heat capacities show phase transition points together with stability changes, which are determined by the variable β . The three geometric frameworks of Weinhold, Ruppeiner and Quevedo consistently demonstrate how systems interact and undergo phase transitions because their curvature properties depend on the value of β . Small entropy values cause η to produce extremely low results, which demonstrate that emissions occur in a very dense state. The study shows that β controlled entropy corrections change the thermodynamic, geometric and radiative characteristics of the Schwarzschild AdS 5 × S 5 black hole which leads to better understanding of its fundamental properties. We also study the test particle dynamics in the background of a five-dimensional Schwarzschild black hole by performing an analysis of the Lyapunov exponent and the corresponding effective force. The study examines how these quantities change with different l p parameter values to determine their stability and force properties in AdS geometry. The results demonstrate that changes in spatial curvature and parameter values affect both chaotic movements and radial motion patterns in black hole spacetime environments.

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