In this paper, we propose and analyze a recovery approach for trilinear finite element approximations on locally-refined hexahedral meshes for a class of elliptic eigenvalue problems. In the approach a local high-order interpolation recovery is followed by some gradient averaging based defect correction scheme. It is proved theoretically and shown numerically that our recovery approach can produce highly accurate eigenpair approximations. And we observe from our numerical experiments that the recovered eigenvalue approximation from the gradient averaging based defect correction approximates the exact eigenvalue from below. Furthermore, this approach has been applied to electronic structure calculations to improve the total energy approximations with small extra overheads.