  The phase and stability of black strings in the Einstein-Gauss-Bonnet (EGB)
theory are investigated by using the large D effective theory approach. The
spacetime metric and thermodynamics are derived up to the next-to-leading order
(NLO) in the 1/D expansion. We find that the entropy current defined by the
Iyer-Wald formula follows the second law. As in the Einstein theory, the
entropy difference from the total mass produces an entropy functional for the
effective theory. Including the NLO correction, we find that for the large
Gauss-Bonnet coupling constant $\alpha_{\rm GB}$, the Gregory-Laflamme
instability of uniform black strings needs longer wavelength. Moreover, we show
that the critical dimension, beyond which non-uiform black strings becomes more
stable than uniform ones, increases as $\alpha_{\rm GB}$ becomes large, and
approaches to a finite value for $\alpha_{\rm GB}\to \infty$.
