  Topological phase transitions occur when the electronic bands change their
topological properties, typically featuring the closing of the bandgap. While
the influence of topological phase transitions on electronic and optical
properties has been extensively studied, its implication on phononic properties
and thermal transport remains unexplored. In this work, we use first-principles
simulations to show that certain phonon modes are significantly softened near
topological phase transitions, leading to increased phonon-phonon scattering
and reduced lattice thermal conductivity. We demonstrate this effect using two
model systems: pressure-induced topological phase transition in $\rm ZrTe_5$
and chemical composition induced topological phase transition in
$\rm{Hg_{1-x}Cd_{x}Te}$. We attribute the phonon softening to emergent Kohn
anomalies associated with the closing of the bandgap. Our study reveals the
strong connection between electronic band structures and lattice instabilities
and opens up a potential direction towards controlling heat conduction in
solids.
