In this work we study a general vector-tensor model of dark energy with a Gauss-Bonnet term coupled to a vector field and without explicit potential terms. Considering a spatially flat FRW type universe and a vector field without spatial components, the cosmological evolution is analysed from the field equations of this model, considering two sets of parameters. In this context, we have shown that it is possible to obtain an accelerated expansion phase of the universe, since the equation state parameter $w$ satisfies the restriction $-1<w<-1/3$ (for suitable values of model parameters). Further, analytical expressions for the Hubble parameter $H$, equation state parameter $w$ and the invariant scalar $\phi$ are obtained. We also find that the square of the speed of sound is negative for all values of redshift, therefore, the model presented here shows a sign of instability under small perturbations. We finally perform an analysis using $H(z)$ observational data and we find that for the free parameter $\xi$ in the interval $(-23.9, -3.46)\times 10^{-5}$, at $99.73\%$ C.L. (and fixing $\eta = -1$ and $\omega = 1/4$), the model has a good fit to the data.