The　emerging　load　prediction　techniques　support　up-front　and　rational　resource　provisioning　in　edge　systems　to　enhance　system　efficiency　and　Quality-of-Service　(QoS).　Classic　prediction　methods　may　handle　loads　with　apparent　trends,　but　they　cannot　achieve　accurate　prediction　for　highly-variable　edge　loads.　With　the　advantage　of　sequential　data　analysis,　recurrent　neural　networks　(RNNs)　are　often　used　for　load　prediction　but　reveal　limited　generalization　ability　and　low　training　efficiency.　Moreover,　it　is　hard　to　obtain　a　well-performed　prediction　model　by　discrete　single-edge　training　with　insufficient　historical　data.　To　address　these　important　challenges,　we　propose　a　novel　Multi-edge　Cooperative　universal　framework　for　load　Prediction　with　Personalized　Federated　deep　learning　(MC-2PF),　enabling　multi-edge　cooperative　training　of　load　prediction　models.　Specifically,　to　solve　the　client-drift　issue　in　federated　learning　(FL)　caused　by　distinct　data　distribution,　we　customize　personalized　models　for　each　edge　by　independent　control　parameters　and　theoretically　analyze　the　model　convergence　improvement.　Meanwhile,　we　prove　the　generalization　bound　of　the　MC-2PF　and　its　universality　to　RNN-based　prediction　models　through　a　practical　example.　Using　the　real-world　testbed　and　load　datasets,　extensive　experiments　verify　the　effectiveness　and　practicality　of　the　MC-2PF　for　different　RNN-based　prediction　models.　Compared　to　state-of-the-art　frameworks,　the　MC-2PF　achieves　higher　prediction　accuracy,　faster　convergence,　and　stronger　adaptiveness.