  We determine the impact of anharmonic thermal vibrations on the fundamental
band gap of 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.
