Stars on tight orbits around the supermassive black hole at the Galactic Center pass through regions where the dark matter (DM) density may be strongly enhanced. We compute the orbit-averaged DM-induced energy exchange for S4714 as an example. It is a star on an exceptionally close and relativistic orbit around Sagittarius A*. For a spiked dark matter profile, the exchange reaches the stellar luminosity at {p} 10^{-36} cm^2 for MeV–GeV masses and {e} 5 ^{-38} cm^2 for sub-MeV masses, opening a new annihilation-free route toward dark-star phases. These cross sections lie within the range predicted by freeze-in scenarios and are consistent with cosmic-ray-boosted and solar-reflection dark matter constraints.