In this paper, we address the fundamental problem of sparse signal recovery for temporally correlated multiple measurement vectors (MMV) in a Bayesian framework. The temporal correlation of the sparse vector is modeled using a first order autoregressive process. In the case of time varying sparse signals, conventional tracking methods like Kalman filtering fail to exploit the sparsity of the underlying signal. Moreover, the computational complexity associated with sparse Bayesian learning (SBL) renders it infeasible even for moderately large datasets. To address this issue, we utilize variational approximation technique (which allows to obtain analytical approximations to the posterior distributions of interest even when exact inference of these distributions is intractable) to propose a novel fast algorithm called space alternating variational estimation with Kalman filtering (SAVE-KF). Similarly as for SAGE (space-alternating generalized expectation maximization) compared to EM, the component-wise approach of VB appears to allow to avoid a lot of bad local optima, explaining the better performance, apart from lower complexity. Simulation results also show that the proposed algorithm has a faster convergence rate and achieves lower mean square error (MSE) than other state of the art fast SBL methods for temporally correlated measurement vetors.