<p>For the tempered nonlocal Laplacian, an efficient and accurate numerical evaluation in multi-dimensions is challenging due to the nature of a singular integral. In this research, we present a new approximation using the generating function or multiplier approximation theory to discretize the multi-dimensional tempered nonlocal Laplacian defined by a hypersingular integral. We show that the approximation can uniformly achieve the convergence of the second order, independent of the operator parameters. Using the proposed approximation, we construct a simple and easy-to-implement finite difference scheme to solve elliptic and parabolic equations with the tempered nonlocal Laplacian. We provide the analysis for convergence and stability of the scheme for the elliptic and parabolic equations. We also present a fast algorithm with quasi-linear complexity of the scheme for computing the tempered operator. Several numerical examples demonstrate the accuracy and efficiency of our algorithm and verify our theory.</p>