  A topological phase transition from a trivial insulator to a $\mathbb{Z}_2$
topological insulator requires the bulk band gap to vanish. In the case of
noncentrosymmetric materials, these phases are separated by a gapless Weyl
semimetal phase. However, at finite temperature, the gap is affected by atomic
motion, through electron-phonon interaction, and by thermal expansion of the
lattice. As a consequence, the phase space of topologically nontrivial phases
is affected by temperature. In this paper, the pressure and temperature
dependence of the indirect band gap of BiTeI is investigated from first
principles. We evaluate the contribution from both electron-phonon interaction
and thermal expansion, and show that their combined effect drives the
topological phase transition towards higher pressures with increasing
temperature. Notably, we find that the sensitivity of both band extrema to
pressure and topology for electron-phonon interaction differs significantly
according to their leading orbital character. Our results indicate that the
Weyl semimetal phase width is increased by temperature, having almost doubled
by 100 K when compared to the static lattice results. Our findings thus provide
a guideline for experimental detection of the nontrivial phases of BiTeI and
illustrate how the phase space of the Weyl semimetal phase in
noncentrosymmetric materials can be significantly affected by temperature.
