  The gradient flow equation in the 2D $O(N)$ nonlinear sigma model with
lattice regularization is solved in the leading order of the $1/N$ expansion.
By using this solution, we analytically compute the thermal expectation value
of a lattice energy--momentum tensor defined through the gradient flow. The
expectation value reproduces thermodynamic quantities obtained by the standard
large-$N$ method. This analysis confirms that the above lattice
energy--momentum tensor restores the correct normalization automatically in the
continuum limit, in a system with a non-perturbative mass gap.
