Anomaly quenching and dynamical cooling of Hawking evaporation in Horndeski gravity
We investigate the semiclassical Hawking evaporation of a two-dimensional Callan-Giddings-Harvey-Strominger (CGHS) black hole perturbed by first-order Horndeski scalar-tensor couplings. By evaluating the exact scalar curvature invariants, we demonstrate that coordinate singularities near the apparent horizon parametrize a physical deformation of the surface gravity while preserving a strictly regular event horizon. In the linear-order $γ_3 ΦX$ coupling sector, the surface gravity and correspondingly the Hawking temperature decreases dynamically as the black hole loses mass. Extrapolating this cooling effect via a resummed temperature equation suggests that evaporation halts, potentially leaving behind a stable, macroscopic cold remnant at the boundary of perturbative control. Utilizing both the conformal trace anomaly and the Robinson-Wilczek gravitational anomaly, we establish that the asymptotic radiation flux identically reflects this geometric shift. Consequently, the fine-grained entanglement entropy of the Hawking radiation departs from linear growth and transitions to an asymptotic logarithmic regime. This indicates that quantum information may be permanently retained within the remnant, altering the standard semiclassical Page curve without requiring a breakdown of horizon regularity.