Black Hole Thermodynamics with a Variable Planck Constant: Generalized Laws, Microscopic Interpretation, and Holographic Structure

Here, I investigate the thermodynamics and information content of black holes in a theoretical framework where both Newton’s constant G and Planck’s constant ħ are promoted to functions of a unification scalar field Π. This modification, motivated by anomaly-induced effective actions and supported by explicit numerical black-hole solutions with scalar hair, leads to a generalized entropy formula S_BH ∝ A * Π_h^3, where Π_h is the value of the field on the horizon. We derive the generalized first law, identify the new thermodynamic potential conjugate to Π_h, and verify the generalized second law for quantum emission processes. We then develop a microscopic interpretation of S_BH ∝ A * Π_h^3 based on (i) the phase–space volume modified by ħ(Π), (ii) a running holographic central charge c_eff(Π_h) ∝ Π_h^3, (iii) a Cardy-like microscopic density of states, and (iv) information compression due to the scalar dependence of the quantum resolution scale. The resulting theory revises several textbook principles and provides a unified picture connecting horizon microphysics, holography, and black-hole evaporation.