  We investigate the role of temperature on the topological insulating state of
metal dichalcogenide monolayers, 1T$^\prime$-MX$_2$ (M=W, Mo and X=S, Se).
Using first principles calculations based on density functional theory, we
consider three temperature-related contributions to the topological band gap:
electrons coupling with short-wavelength phonons, with long-wavelength phonons
\textit{via} Fr\"ohlich coupling, and thermal expansion. We find that
electron-phonon coupling promotes the topology of the electronic structures of
all 1T$^\prime$-MX$_2$ monolayers, while thermal expansion acts as a
counteracting effect. Additionally, we derive the band renormalization from
Fr\"ohlich coupling in the two-dimensional context and observe its relatively
modest contribution to 1T$^\prime$-MX$_2$ monolayers. Finally, we present a
simplified physical picture to understand the "inverse Varshni" effect driven
by band inversion in topological insulators. Our work reveals that, among the
four 1T$^\prime$-MX$_2$ studied monolayers, MoSe$_2$ is a promising candidate
for room temperature applications as its band gap displays remarkable
resilience against thermal expansion, while the topological order of WS$_2$ can
be tuned under the combined influence of strain and temperature. Both materials
represent novel examples of temperature promoted topological insulators.
