  Non-uniform black strings (NUBS) are studied by the large $D$ effective
theory approach. By solving the near-horizon geometry in the $1/D$ expansion,
we obtain the effective equation for the deformed horizon up to the
next-to-next-to-leading order (NNLO) in $1/D$. We also solve the far-zone
geometry by the Newtonian approximation. Matching the near and far zones, the
thermodynamic variables are computed in the $1/D$ expansion. As the result, the
large $D$ analysis gives a critical dimension $D_*\simeq13.5$ at which the
translation-symmetry-breaking phase transition changes between first and second
order. This value of $D_*$ agrees perfectly, within the precision of the $1/D$
expansion, with the result previously obtained by E. Sorkin through the
numerical resolution. We also compare our NNLO results for the thermodynamics
of NUBS to earlier numerical calculations, and find good agreement within the
expected precision.
