  The gravitational-wave event GW170817, together with the electromagnetic
counterpart, shows that the speed of tensor perturbations $c_T$ on the
cosmological background is very close to that of light $c$ for the redshift
$z<0.009$. In generalized Proca theories, the Lagrangians compatible with the
condition $c_T=c$ are constrained to be derivative interactions up to cubic
order, besides those corresponding to intrinsic vector modes. We place
observational constraints on a dark energy model in cubic-order generalized
Proca theories with intrinsic vector modes by running the Markov chain Monte
Carlo (MCMC) code. We use the cross-correlation data of the integrated
Sachs-Wolfe (ISW) signal and galaxy distributions in addition to the data sets
of cosmic microwave background, baryon acoustic oscillations, type Ia
supernovae, local measurements of the Hubble expansion rate, and redshift-space
distortions. We show that, unlike cubic-order scalar-tensor theories, the
existence of intrinsic vector modes allows the possibility for evading the
ISW-galaxy anticorrelation incompatible with the current observational data. As
a result, we find that the dark energy model in cubic-order generalized Proca
theories exhibits a better fit to the data than the cosmological constant, even
by including the ISW-galaxy correlation data in the MCMC analysis.
