  We study the cosmology on the Friedmann-Lemaitre-Robertson-Walker background
in scalar-vector-tensor theories with a broken $U(1)$ gauge symmetry. For
parity-invariant interactions arising in scalar-vector-tensor theories with
second-order equations of motion, we derive conditions for the absence of
ghosts and Laplacian instabilities associated with tensor, vector, and scalar
perturbations at linear order. This general result is applied to the
computation of the primordial tensor power spectrum generated during inflation
as well as to the speed of gravity relevant to dark energy. We also construct a
concrete inflationary model in which a temporal vector component $A_0$
contributes to the dynamics of cosmic acceleration besides a scalar field
$\phi$ through their kinetic mixings. In this model, we show that all the
stability conditions of perturbations can be consistently satisfied during
inflation and subsequent reheating.
