In modified gravity, dynamical Schwarzschild black holes under scalar waves exhibit non-thermal particle creation while preserving the generalized second law and forming stable zero-temperature remnants at the extremal bound.

We investigate the dynamical and thermodynamic evolution of a Schwarzschild black hole in Modified Gravity (MOG) perturbed by a scalar gravitational wave breathing mode. By evaluating the linearized modified Einstein equations at the near-horizon boundary, we reduce the spatial wave operator to a closed-form temporal ordinary differential equation, thereby explicitly deriving the damped-oscillatory kinematics of the scalar strain. Using a quasi-adiabatic approximation, we show that the effective surface gravity and dynamical temperature are linearly modulated by the perturbation amplitude and velocity. These rapid geometric fluctuations break the semiclassical adiabatic regime, triggering explicitly non-thermal particle creation analogous to the dynamical Casimir effect. Furthermore, we resolve a local thermodynamic paradox concerning apparent horizon area fluctuations. We prove that first-order geometric perturbations $\mathcal{O}(h_b)$ are fully reversible kinematic artifacts, whereas irreversible entropy generation is a strictly second-order $\mathcal{O}(h_b^2)$ effect driven by the Raychaudhuri expansion, thereby preserving the Generalized Second Law. Finally, we apply these mechanisms to the black hole information paradox. We show that treating the MOG deformation parameter as a quantum-scale running coupling, $\alpha(M)$, mathematically decouples the effective gravitational charge from linear mass scaling. This dynamically forces the evaporating black hole toward the extremal limit ($M_G \to Q_G$), smoothly quenching the Hawking temperature to zero and yielding a thermodynamically stable, information-preserving remnant.