  We study the approximation of the spectrum of a second-order elliptic
differential operator by the Hybrid High-Order (HHO) method. The HHO method is
formulated using cell and face unknowns which are polynomials of some degree
$k\geq0$. The key idea for the discrete eigenvalue problem is to introduce a
discrete operator where the face unknowns have been eliminated. Using the
abstract theory of spectral approximation of compact operators in Hilbert
spaces, we prove that the eigenvalues converge as $h^{2t}$ and the
eigenfunctions as $h^{t}$ in the $H^1$-seminorm, where $h$ is the mesh-size,
$t\in [s,k+1]$ depends on the smoothness of the eigenfunctions, and $s>\frac12$
results from the elliptic regularity theory. The convergence rates for smooth
eigenfunctions are thus $h^{2k+2}$ for the eigenvalues and $h^{k+1}$ for the
eigenfunctions. Our theoretical findings, which improve recent error estimates
for Hybridizable Discontinuous Galerkin (HDG) methods, are verified on various
numerical examples including smooth and non-smooth eigenfunctions. Moreover, we
observe numerically in one dimension for smooth eigenfunctions that the
eigenvalues superconverge as $h^{2k+4}$ for a specific value of the
stabilization parameter.
