  The asymptotic partition function for quantized Abelian Higgs vortices at
high temperature $T$ is found to leading and subleading order, and from this
the equation of state of the vortex gas is derived, including the first quantum
correction. It is assumed that the Hamiltonian is proportional to the
Laplace--Beltrami operator on the moduli space of static $N$-vortex solutions.
The partition function is calculated using the total volume and total scalar
curvature of the moduli space.
