  The temperature renormalization of the bulk band structure of a topological
crystalline insulator, SnTe, is calculated using first principles methods. We
explicitly include the effect of thermal-expansion-induced modification of
electronic states and their band inversion on electron-phonon interaction. We
show that the direct gap decreases with temperature, as both thermal expansion
and electron-phonon interaction drive SnTe towards the phase transition to a
topologically trivial phase as temperature increases. The band gap
renormalization due to electron-phonon interaction exhibits a non-linear
dependence on temperature as the material approaches the phase transition,
while the lifetimes of the conduction band states near the band edge show a
non-monotonic behavior with temperature. These effects should have important
implications on bulk electronic and thermoelectric transport in SnTe and other
topological insulators.
