Abstract
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 [Formula: see text], they yield the same near-horizon Hawking-temperature rescaling, [Formula: see text]. 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 [Formula: see text] — here [Formula: see text] parametrizes the leading Planck-suppressed deformation of the energy sector ([Formula: see text] term), while [Formula: see text] (with [Formula: see text] the parameters of the spatial-momentum corrections introduced therein) parametrizes that of the spatial-momentum sector ([Formula: see text] term) — and vanishes on the symmetric branch. Furthermore, we examine how the result depends on the choice of [Formula: see text], demonstrating that although the [Formula: see text] 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.