In this article, we generalize the set of manifolds over which the \(L^p\) -spectrum of the Laplacian on k-forms depends on p. We will consider the case of manifolds that are warped products at infinity and certain quotients of Hyperbolic space. In the case of warped products at infinity we prove that the \(L^p\) -spectrum of the Laplacian on k-forms contains a parabolic region which depends on k, p and the limiting curvature \(a_0\) at infinity. For \(M=\mathbb {H}^{N+1}/\Gamma \) with \(\Gamma \) a geometrically finite group such that M has infinite volume and no cusps, we prove that the \(L^p\) -spectrum of the Laplacian on k-forms is a exactly a parabolic region together with a set of isolated eigenvalues on the real line.
