While 360° videos are gaining popularity due to the emergence of VR technologies, storing and streaming such videos can incur up to 20X higher overheads than traditional HD content. Edge caching, which involves caching and serving 360° videos from edge servers, is one possible approach for addressing these overheads. Prior work on 360° video caching has been based on using past history to cache tiles that are likely to be in a viewer's field of view and has not considered methods to intelligently share a limited edge cache across a set of videos that exhibit large variations in their popularity, size, content, and user abandonment patterns. Towards this end, we present CAVE, an adaptive edge caching framework that intelligently optimizes cache allocation across a set of videos taking into account video content, size, and popularity. Our experiments using realistic video workloads shows CAVE improves cache hit-rates, and thus network saving, by up to 50% over state-of-the-art approaches, while also scaling to up to two thousand videos per edge cache. In addition, in terms of scalability, our developed algorithm is embarrassingly parallel, allowing CAVE to scale beyond state-of-the-art solutions that typically do not support parallelization.