  We employ semiclassical quantization to calculate spectrum of quantum KdV
charges in the limit of large central charge $c$. Classically, KdV charges
$Q_{2n-1}$ generate completely integrable dynamics on the co-adjoint orbit of
the Virasoro algebra. They can be expressed in terms of action variables $I_k$,
e.g.~as a power series expansion. Quantum-mechanically this series becomes the
expansion in $1/c$, while action variables become integer-valued quantum
numbers $n_i$. Crucially, classical expression, which is homogeneous in $I_k$,
acquires quantum corrections that include terms of subleading powers in $n_k$.
At first two non-trivial orders in $1/c$ expansion these ``quantum'' terms can
be fixed from the analytic form of $Q_{2n-1}$ acting on the primary states. In
this way we find explicit expression for the spectrum of $Q_{2n-1}$ up to first
three orders in $1/c$ expansion. We apply this result to study thermal
expectation values of $Q_{2n-1}$ and free energy of the KdV Generalized Gibbs
Ensemble.
