Doubly Special Relativity (DSR) introduces an invariant energy scale; however, in curved spacetime, the physical meaning of the deformation energy remains ambiguous. In this work, we study static, spherically symmetric black holes by comparing two widely used implementations of DSR effects: local-frame modified dispersion relations on a fixed background and rainbow metrics. We demonstrate that, when both descriptions are evaluated at the same operational energy scale E ★ , they yield the same near-horizon Hawking-temperature rescaling, T(E ★ ) = T 0 g/f. We subsequently apply this correspondence to the Amelino-Camelia and Magueijo-Smolin models, as well as to the generalized two-parameter DSR family of Jafari and Good, where the leading correction depends only on Δα–α 2 —here a2 parametrizes the leading Planck-suppressed deformation of the energy sector (E 2 term), while Δα ≡ α 3 -α 1 (with α 1 , α 3 the parameters of the spatial-momentum corrections introduced therein) parametrizes that of the spatial-momentum sector (p 2 term)— and vanishes on the symmetric branch. Furthermore, we examine how the result depends on the choice of E ★ , demonstrating that although the g/f structure is fixed once a prescription is adopted, the magnitude and mass dependence of the correction remain prescription dependent. Finally, we discuss illustrative implications for phenomenological bounds and black-hole evaporation. Accordingly, our result is best interpreted as a conditional equivalence between two curved-spacetime implementations of DSR, rather than as a universal first-principles prediction.
Boumali et al. (Fri,) studied this question.