  Monolayer 1T'-WTe$_2$ has been the first two-dimensional crystal where a
quantum spin Hall phase was experimentally observed. In addition, recent
experiments and theoretical modeling reported the presence of a robust
excitonic insulating phase. While first-principles calculations with hybrid
functionals and several measurements at low temperatures suggest the presence
of a band gap of the order of 50 meV, experiments could confirm the presence of
the helical edge states only up to 100 K. Here, we study with first-principle
simulations the temperature effects on the electronic structure of monolayer
1T'-WTe$_2$ and consider the contributions of both thermal expansion and
electron-phonon coupling. First, we show that thermal expansion is weak but
tends to increase the indirect band gap. Then, we calculate the effect of
electron-phonon coupling on the band structure with non-perturbative methods
and observe a small reduction of the band inversion with increasing
temperature. Notably, the topological phase and the presence of a finite gap
are found to be particularly robust to thermal effects up to and above room
temperature.
