Thermodynamic and Observational Implications of Black Holes in Toroidal Geometry

Abstract We investigate the thermodynamic and observational implications for charged torus-like black holes, a class of solutions distinct from classical Schwarzschild black holes. We explicitly derive the fundamental thermodynamic properties, such as heat capacity, pressure–volume diagram, isothermal compressibility, Helmholtz free energy, and Gibbs free energy, under different entropy models. We find that only the exponential corrected entropy demonstrates multiple phase transitions, which we validate with the Ricci scalar divergence obtained from the Ruppeiner formalism. This indicates that the exponential corrected entropy is more sensitive to the black hole microstructure as compared to the Hawking–Bekenstein and Rényi entropy models within the nonspherical (toroidal) horizon. In addition, we study the sparsity and emission rates of Hawking radiation, demonstrating that exponential correction entropy yields more consistent and stable behavior. In our observational analysis, we graphically demonstrate the behavior of redshift, blueshift, and gravitational shift, and identify specific conditions where the photon sphere radius exceeds the innermost stable circular orbit radius, which depends on the values of parameters such as electric charge and cosmological constant. The novel insight of this work is that, despite this violation, our computed redshift, blueshift, and gravitational shifts fall within the range of the observational data of NGC 4258 and UGC 3789.

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