The density of states $g(\epsilon)$ for a spin-$S$ gas with spin degeneracy $g_S$ in a $D$-dimensional box with energy relation $\epsilon(\mathbf{p})$ is
\[g(\epsilon) = g_s\left(\frac{L}{2\pi\hbar}\right)^D \int{d^Dp\delta(\epsilon - \epsilon(p))} = g_s\left(\frac{L}{2\pi\hbar}\right)^D \int d\Omega_D \frac{p(\epsilon)^{D-1}}{\left|\frac{\partial \epsilon}{\partial p}(p(\epsilon))\right|}\]