  We study the effects of temperature on the band structure of the Bi$_2$Se$_3$
family of topological insulators using first-principles methods. Increasing
temperature drives these materials towards the normal state, with similar
contributions from thermal expansion and from electron-phonon coupling. The
band gap changes with temperature reach $0.3$ eV at $600$ K, of similar size to
the changes caused by electron correlation. Our results suggest that
temperature-induced topological phase transitions should be observable near
critical points of other external parameters.
