  We evaluate the high temperature limit of the free energy of spin glasses on
the hypercube with Hamiltonian $H_N(\sigma) = \sigma^T J \sigma$, where the
coupling matrix $J$ is drawn from certain symmetric orthogonally invariant
ensembles. Our derivation relates the annealed free energy of these models to a
spherical integral, and expresses the limit of the free energy in terms of the
limiting spectral measure of the coupling matrix $J$. As an application, we
derive the limiting free energy of the Random Orthogonal Model (ROM) at high
temperatures, which confirms non-rigorous calculations of Marinari et al.
(1994). Our methods also apply to other well-known models of disordered
systems, including the SK and Gaussian Hopfield models.
