Black-hole thermodynamics in doubly special relativity: near-horizon g/f temperature scaling under a shared operational scale

Doubly Special Relativity (DSR) deforms special-relativistic kinematics by introducing an invariant Planck energy scale $E_{\mathrm{P}\mathrm{l}}$ alongside the speed of light, while preserving the relativity principle. A key issue in curved spacetimes, particularly black-hole thermodynamics, is the operational meaning of the ``energy’’ in modified dispersion relations (MDRs). We compare two common implementations in a controlled static black-hole spacetime: (i) MDRs in local orthonormal frames on a fixed background geometry, and (ii) the rainbow-metric approach with an energy-dependent family of effective metrics. For static, spherically symmetric horizons and using a consistent finite operational energy scale $E_{\star }$ for emitted quanta, both yield the same near-horizon temperature rescaling $$T(E_{\star })=T_0\frac{g(E_{\star }\mathrm{/}{E}_{\mathrm{P}\mathrm{l}})}{f(E_{\star }\mathrm{/}{E}_{\mathrm{P}\mathrm{l}})},T_0={\kappa }_0\mathrm{/}(2\pi ),$$ where $f$ and $g$ are the standard rainbow/MDR functions. This establishes a universality of the tunneling/surface-gravity temperature, with deformation entering solely via the ratio $g\mathrm{/}f$. We illustrate for Amelino-Camelia MDR and Magueijo-Smolin DSR (where $f=g$, implying $T(E_{\star })=T_0$). Extending to a two-parameter generalized DSR (G-DSR) with leading parameters $({\alpha }_2,\mathrm{\Delta }\alpha )$, we obtain $$T_{\mathrm{G}\mathrm{D}\mathrm{R}\mathrm{S}}(E_{\star })=T_0\sqrt{\frac{1-2\mathrm{\Delta }\alpha (E_{\star }\mathrm{/}{E}_{\mathrm{P}\mathrm{l}})}{1-2{\alpha }_2(E_{\star }\mathrm{/}{E}_{\mathrm{P}\mathrm{l}})}}\simeq T_0[1-(\mathrm{\Delta }\alpha -{\alpha }_2)E_{\star }\mathrm{/}{E}_{\mathrm{P}\mathrm{l}}]\mathrm{.}$$ We discuss the role of $\mathrm{\Delta }\alpha -{\alpha }_2$ (vanishing correction for the symmetric $\mathrm{\Delta }\alpha ={\alpha }_2$ subfamily) and note that further model dependence arises from phase-space measures, greybody factors, and non-linear composition laws. Corrections are strongly suppressed for macroscopic black holes and become relevant only near the Planck regime.