A gas of N Bogomol'nyi vortices in the Abelian Higgs model is studied on a
compact Riemann surface of genus g and area A. The volume of the moduli
space is computed and found to depend on N,g and A, but not on other
details of the shape of the surface. The volume is then used to find the
thermodynamic partition function and it is shown that the thermodynamical
properties of such a gas do not depend on the genus of the Riemann surface.Comment: LaTex file, 17 pages. To appear in Comm. Math. Phy