Ameer HamzaCIIT/SP24-RMT-007/LHRDr. Abdul JawadLHR TP 100702026-05-132025https://repository.cuilahore.edu.pk/123456789/3902In this thesis, we investigate charged black hole through the framework of Bekenstein- Hawking entropy and modified exponential entropy. These entropies are selected for their direct proportionality to the black hole’s horizon area and its natural derivation from Ein- stein’s field equations. The Bekenstein–Hawking entropy is a universally accepted and geometrically well-defined measure that is directly proportional to the area of the event horizon. On the other hand, the exponential entropy enhances the classical framework by integrating non-extensive and correlation effects inherent to gravitational systems, facili- tating regulated deviations from conventional thermodynamics. These two entropies offer both theoretical precision and practical flexibility in the analysis of black hole thermody- namics. We consider the Davies phase transition formalism, and discuss the stability of a black hole in its respective space and calculate its topological properties for dynamic stability. The expressions for mass, temperature, specific heat and divergence points are calculated. Also, we calculate the winding number an total topological charge with the help of phase portraits to discuss the local stable behavior of under consideration black hole for both entropy expression.enDepartment of MathematiccsSP24Mathematiccslack hole thermodynamicsModified entropyPhase transitionsDr. Abdul JawadThesis