  We determine the topological phase diagram of
BiTl(S$_{1-\delta}$Se$_{\delta}$)$_2$ as a function of doping and temperature
from first-principles calculations. Due to electron\textendash phonon
interaction, the bands are renormalized at finite temperature, allowing for a
transition between the trivial ($Z_2=0$) and non-trivial ($Z_2=1$) topological
phase. We find two distinct regions of the phase diagram with non-trivial
topology. In BiTlS$_2$, the phonons promote the crystal to the topological
phase at high temperature, while in BiTlSe$_2$, the topological phase exists
only at low temperature. This behaviour is explained by the symmetry of the
phonon coupling potential, whereby the even phonon modes (whose potential is
even under inversion) promote the topological phase and the odd phonon modes
promote the trivial phase.
