Cyber-physical power systems (CPPS) integrate information and communication technology into conventional electric power systems to facilitate bidirectional communication of information and electric power between users and power grids. Despite its benefits, the open communication environment of CPPS is vulnerable to various security attacks. This paper proposes a federated deep learning-based architecture to detect false data injection attacks (FDIAs) in CPPS. The proposed work offers a strong, decentralized alternative with the ability to boost detection accuracy while maintaining data privacy, presenting a significant opportunity for real-world applications in the smart grid. This framework combines state-of-the-art machine learning and deep learning models, which are used in both centralized and federated learning configurations, to boost the detection of false data injection attacks in cyber-physical power systems. In particular, the research uses a multi-stage detection framework that combines several models, including classic machine learning classifiers like Random Forest and ExtraTrees Classifiers, and deep learning architectures such as Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU). The results demonstrate that Bidirectional GRU and LSTM models with attention layers in a federated learning setup achieve superior performance, with accuracy approaching 99.8%. This approach enhances both detection accuracy and data privacy, offering a robust solution for FDIA detection in real-world smart grid applications.