  We study the temperature dependence of the Rashba-split bands in the bismuth
tellurohalides BiTe$X$ $(X=$ I, Br, Cl) from first principles. We find that
increasing temperature reduces the Rashba splitting, with the largest effect
observed in BiTeI with a reduction of the Rashba parameter of $40$% when
temperature increases from $0$ K to $300$ K. These results highlight the
inadequacy of previous interpretations of the observed Rashba splitting in
terms of static-lattice calculations alone. Notably, we find the opposite
trend, a strengthening of the Rashba splitting with rising temperature, in the
pressure-stabilized topological-insulator phase of BiTeI. We propose that the
opposite trends with temperature on either side of the topological phase
transition could be an experimental signature for identifying it. The predicted
temperature dependence is consistent with optical conductivity measurements,
and should also be observable using photoemission spectroscopy, which could
provide further insights into the nature of spin splitting and topology in the
bismuth tellurohalides.
