  We consider a perturbative Gauss-Bonnet term supplementing the
Einstein-Hilbert action, and evaluate its effect on the spectrum of the scalar
mode that triggers the Gregory-Laflamme instability of black strings in five
dimensional General Relativity. After studying some properties of the static
black string, we provide the correction to the Lichnerowicz operator up to
$\mathcal{O}(\alpha^2)$. For the scalar mode of the gravitational perturbation,
we find a master variable and study its spectrum, providing an analysis of the
regime of validity of our scheme. We show that the instability persists under
the inclusion of the $R^2$-correction, and that the critical wavelength
increases with the value of $\alpha/r_+^2<<1$, providing new evidence of the
phenomenon which was already observed in other approaches. We also construct
the boosted black strings and compute the correction to the mass, the momentum
and the tension due to the higher curvature term. The presence of the
dimensionful coupling $\alpha$ spoils the validity of the Smarr relation, which
is the gravitational version of the Euler relation which must hold for every
homogeneous, thermodynamic system. We show that this identity can be restored
by working in an extended thermodynamic setup that includes variations of the
Gauss-Bonnet coupling.
