Recently, we proposed a weak Galerkin finite element method for the Laplace eigenvalue problem. In this paper, we present two-grid and two-space skills to accelerate the weak Galerkin method. By choosing parameters properly, the two-grid and two-space weak Galerkin method not only doubles the convergence rate, but also maintains the asymptotic lower bounds property of the weak Galerkin method. Some numerical examples are provided to validate our theoretical analysis.The eigenvalue problem arises from many branches of mathematics and physics, including quantum mechanics, fluid mechanics, stochastic process and structural mechanics. A variety of applications of eigenvalue problem, especially the Laplacian eigenvalue problem, are surveyed by a recent SIAM review paper [11]. Many numerical methods have been developed for the Laplacian eigenvalue problem, such as finite difference methods [30] and finite element methods [2, 3, 6].The finite element method is one of the efficient approaches for the Laplacian eigenvalue problem for its simplicity and adaptivity on triangular meshes. Due to the minimum-maximum principle, the conforming finite element method always gives the upper bounds for the Laplacian eigenvalues. In order to get accurate intervals for eigenvalues, it is necessary to have lower bounds of eigenvalues. There are mainly two ways, the post-processing method [5,15,17,18,17,26,27] and the nonconforming finite element method [1,13,19,39]. Some specific nonconforming finite element methods provide asymptotic lower bounds for eigenvalues without solving an auxiliary problem, while it seems difficult to construct a high order nonconforming element.