Contrary to previous two-band model studies which find increasing temperature would induce a topological phase transition, we show here through first-principles calculations that the opposite is also realizable, depending on the material's full band structure and symmetry of the electron-phonon coupling potential. This finding explains recent experimental results by Wojek et al. [Nat. Commun. 6, 8463 (2015)NCAOBW2041-172310.1038/ncomms9463]. We show that the topological phase diagram of BiTl(S_{1-δ}Se_{δ})_{2} as a function of doping and temperature contains two distinct regions with nontrivial topology. In BiTlS_{2}, the phonons promote the topological phase at high temperature, while in BiTlSe_{2}, the system is driven back into the trivial phase.