  In scalar-vector-tensor theories with $U(1)$ gauge invariance, it was
recently shown that there exists a new type of hairy black hole (BH) solutions
induced by a cubic-order scalar-vector interaction. In this paper, we derive
conditions for the absence of ghosts and Laplacian instabilities against
odd-parity perturbations on a static and spherically symmetric background for
most general $U(1)$ gauge-invariant scalar-vector-tensor theories with
second-order equations of motion. We apply those conditions to hairy BH
solutions arising from the cubic-order coupling and show that the odd-parity
stability in the gravity sector is always ensured outside the event horizon
with the speed of gravity equivalent to that of light. We also study the case
in which quartic-order interactions are present in addition to the cubic
coupling and obtain conditions under which black holes are stable against
odd-parity perturbations.
