  The presence of spin-polarized charge carriers in metallic magnets provides a
mechanism for spin-lattice interactions mediated by electron-phonon coupling.
Here, we present a theory of this mechanism used to estimate its effect on the
exchange interactions in 2D magnets. Starting from a square lattice model at
half filling, we show that the presence of electron-phonon coupling with
equilibrium phonon distribution leads to a notable suppression of exchange
interactions with temperature. We then apply our approach to the prototypical
2D metallic ferromagnet, Fe$_3$GeTe$_2$, with moderate electron-phonon
coupling. We find that the exchange interactions undergo a renormalization,
leading to a softening of the magnon modes, and suppression of the Curie
temperature by $\sim$10\%. We expect that this effect can be further enhanced
in systems with strong electron-phonon coupling, as well as for non-equilibrium
distribution of phonons induced by strong laser fields or charge currents.
