Medium-to-long-term underground heat recovery systems often exhibit cumulative thermal imbalance that is not adequately described by classical local diffusion equations. This study develops a first-principles framework that links thermal engineering with non-equilibrium statistical physics. We derive a hybrid evolutionary generator that unifies local Gaussian diffusion and non-local fractional Lévy-type transport, enabling representation of cross-cycle memory and long-range correlation. Within the Onsager variational and Martin-Siggia-Rose (MSR) formalisms, cyclic thermal evolution is formulated as a gradient-flow process coupled with a thermodynamic conjugate information field. We further show that gradient phase change materials (PCMs) can modulate generator parameters toward a near scale-invariant regime associated with improved long-term stability. Based on this field structure, a path-integral adjoint optimal-control framework is established for periodic external heat-source operation. The proposed framework provides a physically consistent explanation for long-term thermal fading and a practical theoretical basis for sustainable underground heat recovery.