Fractional-Action Gravity with a Dynamical Parameter: Implications for Accelerated Expansion, Finkelstein-Eddington Black-Hole Geometry, Mass Evolution, and Hawking Temperature

This study investigates gravitational dynamics emerging from fractional-action models in which the weight parameter 𝛼(t) is treated first as a prescribed time-dependent function and subsequently as a fully dynamical scalar field. In the non-dynamical case, the modified action introduces explicit energy nonconservation, leading naturally to Finkelstein-Eddington black-hole geometries with a time-dependent mass function and corresponding variations in horizon radius and Hawking temperature. When 𝛼(t) is promoted to a dynamical field, the theory acquires a consistent conservation structure in which the scalar contributes to the stress-energy budget, mediates energy exchange, and regularizes the mass flow driving M(v). These effects significantly impact black-hole evaporation rates by adding geometric fluxes that can either enhance or suppress quantum mass loss. On cosmological scales, both formulations modify the expansion history: the non-dynamical version produces effective matter nonconservation capable of mimicking dark-energy-like behavior, while the dynamical version supplies a controlled scalar degree of freedom that can drive or regulate cosmic acceleration. Together, these two complementary treatments show how fractional-action gravity reshapes black-hole evolution, Hawking thermodynamics, and cosmological expansion, offering a unified framework with potentially observable deviations from general relativity.