Motivated by classical works of Schechtman and Schmuckenschläger on intersections of $\ell_p$-balls and recent ones in information-based complexity relating random sections of ellipsoids and the quality of random information in approximation problems, we study the threshold behavior of the asymptotic volume of intersections of generalized $p$-ellipsoids. The non-critical behavior is determined under a spectral flatness (Wiener entropy) condition on the semi-axes. In order to understand the critical case at the threshold, we prove a central limit theorem for $q$-norms of points sampled uniformly at random from a $p$-ellipsoid, which is obtained under Noether's condition on the semi-axes.