The Back Projection (BP) algorithm is a very important time-domain methodology for Synthetic Aperture Radar (SAR) imaging. However, conventional autofocus techniques are based on frequency-domain imaging algorithms, and can not be directly applied to BP imagery for error phase estimation. In this paper, an autofocus algorithm for BP imagery is proposed. The algorithm takes image sharpness as an objective function, and employs the coordinate descent optimization scheme to obtain the optimum phase-corrected variables by iterations. In the implementation, with a Legendre approximation of the objective function, the optimal phase estimation can be found analytically for each parameter within an iteration, avoiding computationally expensive line-search procedures. The experimental results with both simulated and measured data confirm the accuracy and effectiveness of the proposed algorithm.