We report on the ultraviolet optical response of a color center in hexagonal boron nitride. We demonstrate a mapping between the vibronic spectrum of the color center and the phonon dispersion in hexagonal boron nitride, with a striking suppression of the phonon assisted emission signal at the energy of the phonon gap. By means of nonperturbative calculations of the electron-phonon interaction in a strongly anisotropic phonon dispersion, we reach a quantitative interpretation of the acoustic phonon sidebands from cryogenic temperatures up to room temperature. Our analysis provides an original method for estimating the spatial extension of the electronic wave function in a point defect.