Minimum entropy autofocus (MEA) has been applied in unmanned aerial vehicle (UAV) synthetic aperture radar (SAR) imagery for its robustness in different circumstances. However, large amount of range cell samples to calculate the gradient for the minimum entropy optimization keeps its optimal convergence, which usually degrades the efficiency in real UAV SAR applications. In this letter, accelerated minimum entropy autofocus is proposed, which leverages both high computational efficiency and phase error estimation precision simultaneously. A strategy of stochastic gradient (SG) calculation is introduced in the MEA optimization with randomly selecting samples in each iteration through a probability distribution function (PDF). Experimental results with real UAV SAR data have validated the superior performance of the proposed SG-MEA algorithm.