  One of the key issues in the physics of topological insulators is whether the
topologically non-trivial properties survive at finite temperatures and, if so,
whether they disappear only at the temperature of topological gap closing.
Here, we study this problem, using quantum fidelity as a measure, by means of
\emph{ab-initio} methods supplemented by an effective dissipative theory built
on the top of the \emph{ab-initio} electron and phonon band structures. In the
case of SnTe, the prototypical crystal topological insulator, we reveal the
presence of a characteristic temperature, much lower than the gap-closing one,
that marks a loss of coherence of the topological state. The transition is not
present in a purely electronic system but it appears once we invoke coupling
with a dissipative bosonic bath. Features in the dependence with temperature of
the fidelity susceptibility can be related to changes in the band curvature,
but signatures of a topological phase transition appear in the fidelity only
though the non-adiabatic coupling with soft phonons. Our argument is valid for
valley topological insulators, but in principle can be generalized to the
broader class of topological insulators which host any symmetry-breaking boson.
