Deep learning models have shown remarkable performance in medical video object segmentation. However, addressing the cross-center domain issue is crucial for achieving consistent performance across different medical facilities. Emerging Source-Free Active Domain Adaptation (SFADA) techniques can enhance the performance of target domain segmentation models, ensuring data privacy and security. While current approaches primarily focus on image-level tasks and mainly emphasize intra-frame pixel correlations, they overlook temporal correlations, which restricts their performance in video frame recommendation. Consequently, this paper proposes the first video-level SFADA method and evaluates it on video polyp segmentation across different data centers. Specifically, the Spatial-Temporal Active Recommendation (STAR) strategy is devised to recommend a few highly valuable frames for annotation by comprehensively evaluating the object spatial correlation and temporal movement density across different video frames, along with a Passive Phase Correction (PPC) module is proposed to suppress the noisy source disruptions of the remaining unlabeled data during the fine-tuning stage. Experimental results demonstrate that with a tiny quantity of annotation, our method significantly improves performance over the lower bound and achieves better performance than existing SOTA methods, which is valuable for practical clinical employment (link) .