For abstractive text summarization, laborious data annotation and time-consuming model training become two high walls, hindering its further progress. Active Learning, selecting a few informative instances for annotation and model training, sheds light on solving these issues. However, only few active learning-based studies focus on abstractive text summarization and suffer from low stability, effectiveness, and efficiency. To solve the problems, we propose a novel LLM-determined curriculum active learning framework. Firstly, we design a prompt to ask large language models to rate the difficulty of instances, which guides the model to train on from easier to harder instances. Secondly, we design a novel active learning strategy, i.e., Certainty Gain Maximization, enabling to select instances whose distribution aligns well with the overall distribution. Experiments show our method can improve stability, effectiveness, and efficiency of abstractive text summarization backbones.