Thermodynamic Free Energy of Spacetime and Entropy-Driven Cosmogenesis

We construct a self-consistent geometric framework linking horizon thermodynamics witha Landau-type phase transition model within a non-singular black hole interior. Utilizing Ja-cobson’s horizon thermodynamics, the macroscopic free energy of a finite-volume topologicalcore U is expressed through curvature energy work integrals, effective horizon temperature,and area-proportional boundary entropy. By examining an axisymmetric quadrupole per-turbation of the core boundary governed by the spherical harmonic Y20(θ), we formallydemonstrate that the linear geometric contributions vanish over the closed manifold. Theleading-order variations of the effective horizon area are quadratic in the order parameter(∼ τ 2). Assuming analyticity and reflection symmetry (τ → −τ ), we derive the Ginzburg-Landau free energy expansion parameters directly from the differential geometry of thedeformed boundary. Under critical thresholds, the quadratic coefficient undergoes a for-mal stability inversion (B < 0), signaling a topological phase transition. Finally, we showthat this geometric deformation perturbs the spatial Laplacian operator, breaking the m-degeneracy of its eigenvalues, which provides a plausible structural template for the analysisof large-scale cosmological anisotropies.