Existing synthetic aperture radar automatic target recognition (SAR ATR) methods have been effective for the classification of seen target classes. However, it is more meaningful and challenging to distinguish the unseen target classes, i.e., open set recognition (OSR) problem, which is an urgent problem for the practical SAR ATR. The key solution of OSR is to effectively establish the exclusiveness of feature distribution of known classes. In this letter, we propose an entropy-awareness meta-learning method that improves the exclusiveness of feature distribution of known classes which means our method is effective for not only classifying the seen classes but also encountering the unseen other classes. Through meta-learning tasks, the proposed method learns to construct a feature space of the dynamic-assigned known classes. This feature space is required by the tasks to reject all other classes not belonging to the known classes. At the same time, the proposed entropy-awareness loss helps the model to enhance the feature space with effective and robust discrimination between the known and unknown classes. Therefore, our method can construct a dynamic feature space with discrimination between the known and unknown classes to simultaneously classify the dynamic-assigned known classes and reject the unknown classes. Experiments conducted on the moving and stationary target acquisition and recognition (MSTAR) dataset have shown the effectiveness of our method for SAR OSR.