We determine the impact of anharmonic thermal vibrations on the fundamental band gap of ${\mathrm{CsPbBr}}_{3}$, a prototypical model system for the broader class of halide perovskite semiconductors. Through first-principles molecular dynamics and stochastic calculations, we find that anharmonic fluctuations are a key effect in the electronic structure of these materials. We present experimental and theoretical evidence that important characteristics, such as a mildly changing band-gap value across a temperature range that includes phase transitions cannot be explained by harmonic phonons thermally perturbing an average crystal structure and symmetry. Instead, the thermal characteristics of the electronic structure are microscopically connected to anharmonic vibrational contributions to the band gap that reach a fairly large magnitude of 450 meV at 425 K.