New Improved Schwarzschild Black Hole and Its Thermodynamics and Topological Classification

We construct a renormalization-group improved Schwarzschild-like black hole geometry using the exact new scheme running for the Newton coupling. The scale identification is implemented via a standard interpolating proper-distance function that smoothly connects the ultraviolet and infrared regimes. We present the resulting coordinate-dependent coupling and the improved metric function, analyzing its asymptotic expansions. The large-distance limit is shown to recover the classical Schwarzschild solution, while the short-distance behavior exhibits a regular de Sitter-like core, demonstrating the regularization of the central singularity. We also analyze the thermodynamic properties of the solution, showing that quantum corrections significantly modify the small-radius behavior, leading to a remnant configuration and a nontrivial phase structure. Finally, we perform a topological classification of the thermodynamic phase space and demonstrate that asymptotically safe effects shift the critical point while preserving the global topological number of the Schwarzschild solution.