  We study a $T\bar{T}$-deformed $O(N)$ vector model, which is classically
equivalent to the Nambu-Goto action with static gauge. The thermal free energy
density can be computed exactly by using the Burgers equation as a special
property of $T\bar{T}$-deformation. The resulting expression is valid for an
arbitrary value of $N$. One may consider a large $N$ limit while preserving
this expression. We try to derive this result in the field-theoretical approach
directly by employing the large $N$ limit. As a result, the leading
contribution coincides with the exact one. That is, the $1/N$ corrections are
cancelled out through a non-trivial mechanism.
