  We study graphene antidot lattices -- superlattices of perforations
(antidots) in a graphene sheet -- using a model that accounts for the
phonon-modulation of the $\pi$-electron hopping integrals. We calculate the
phonon spectra of selected antidot lattices using two different semi-empirical
methods. Based on the adopted model, we quantify the nature of charge carriers
in the system by computing the quasiparticle weight due to the electron-phonon
interaction for an excess electron in the conduction band. We find a very
strong phonon-induced renormalization, with the effective electron masses
exhibiting nonmonotonic dependence on the superlattice period for a given
antidot diameter. Our study provides an indication of polaronic behavior and
points to the necessity of taking into account the inelastic degrees of freedom
in future studies of transport in graphene antidot lattices.
