Thermodynamic Free Energy of Spacetime and Entropy-Driven Cosmogenesis

This work presents a rigorous, non-singular framework for cosmogenesis derived directly from first principles using Jacobson’s horizon thermodynamics. We model the interior of black holes by replacing the classical curvature singularity with a finite-volume four-dimensional topological core U. By establishing explicit classical parameterizations for the core's geometry under axisymmetric tidal deformations, we analytically derive the canonical Ginzburg-Landau (Landau-Khalatnikov) free energy expansion functional F(τ) without relying on phenomenological assumptions or speculative quantum gravity fields. We model the dynamical relaxation of the metric order parameter τ(t) to map the spinodal decomposition that triggers the transition into an internal FRW cosmology, establishing cosmogenesis as an entropy-maximizing phase transition. Furthermore, we calculate the gauge-invariant gravitational wave emission from this metric re-configuration. The resulting spectral energy density of the relic stochastic gravitational-wave background, Ω_GW(f), is shown to strictly converge to an analytical f^3 power-law in the infra-red limit due to causal white-noise constraints, while spatial filtering over the macroscopic and microscopic core boundaries yields a distinctive double-peak profile overlapping with the active sensitivity bands of Pulsar Timing Arrays (PTA/NANOGrav) and space-based interferometers (LISA/DECIGO). Finally, we show that the SO(3) -> SO(2) spatial symmetry breaking at the transition surface induces a predictable quadrupole directional anisotropy in the GW strain. This maps directly to observed large-scale CMB anomalies (such as the quadrupole suppression and the Axis of Evil), providing a clean, фальсифицируемый (testable) cross-correlation signature for upcoming observational missions.