  We develop an approximate theory of phonon-induced topological insulation in
Dirac materials. In the weak coupling regime, long wavelength phonons may favor
topological phases in Dirac insulators with direct and narrow bandgaps. This
phenomenon originates from electron-phonon matrix elements, which change
qualitatively under a band inversion. A similar mechanism applies to weak
Coulomb interactions and spin-independent disorder; however, the influence of
these on band topology is largely independent of temperature. As applications
of the theory, we evaluate the temperature-dependence of the critical thickness
and the critical stoichiometric ratio for the topological transition in
CdTe/HgTe quantum wells and in BiTl(S$_{1-\delta}$Se$_{\delta})_2$,
respectively.
