  The existence of an acoustic plasmon in extrinsic (doped or gated) monolayer
graphene was found recently in an {\it ab initio} calculation with the frozen
lattice [M. Pisarra {\it et al.}, arXiv:1306.6273, 2013]. By the {\em fully
dynamic} density-functional perturbation theory approach, we demonstrate a
strong coupling of the acoustic plasmonic mode to lattice vibrations. Thereby,
the acoustic plasmon in graphene does not exist as an isolated excitation, but
it is rather bound into a combined plasmon-phonon mode. We show that the
coupling provides a mechanism for the {\em bidirectional} energy exchange
between the electronic and the ionic subsystems with fundamentally, as well as
practically, important implications for the lattice cooling and heating by
electrons in graphene.
