  We generalize Feichtinger and Kaiblinger's theorem on linear deformations of
uniform Gabor frames to the setting of a locally compact abelian group $G$.
More precisely, we show that Gabor frames over lattices in the time-frequency
plane of $G$ with windows in the Feichtinger algebra are stable under small
deformations of the lattice by an automorphism of $G \times \widehat{G}$. The
topology we use on the automorphisms is the Braconnier topology. We
characterize the groups in which the Balian--Low theorem for the Feichtinger
algebra holds as exactly the groups with noncompact identity component. This
generalizes a theorem of Kaniuth and Kutyniok on the zeros of the Zak transform
on locally compact abelian groups. We apply our results to a class of
number-theoretic groups, including the adele group associated to a global
field.
