Multicast routing with multiple Quality-of-Service (QoS) constraints has been realized as an promising technology to meet the increasing bandwidth demand of multimedia services. Developing cost-effective multicast routing algorithm plays a crucial role in implementing multicast routing protocol successfully. However, most of prior multicast routing algorithms are dedicated to two QoS constraints, few researches focus on the case with three or more constraints. In this paper, we explore the multicast routing problem with K constraints where K > 2. To address this problem, we propose FAMOUS, a fast multi-constrained multicast routing algorithm. The algorithm leverages an entropy-based process to aggregate all weights into a comprehensive metric for simplifying the problem, and then applies this metric to search a multicast tree on the constructing shortest path tree basis. We conduct complexity analysis and extensive simulations for the proposed algorithm. Both analytical and experimental results demonstrate that FAMOUS is superior to a representative multi-constrained multicast routing algorithm in terms of both speed and accuracy, and thus it is applicable to realistic networking scenarios.