To navigate the carbon emission and functional challenges associated with edge caching within heterogeneous Industrial Internet of Things (IIoT) spanning energy use, cache hit rate, and bandwidth usage, this paper proposes a novel Game Theory Empowered Carbon-Intelligent Federated Multi-Edge Caching framework (GT-FMC). The proposed framework enables distributed collaborative caching by intelligently coordinating edge nodes to optimize content decisions while efficiently integrating content providers (CPs), edge nodes, and users with energy-aware strategies. In GT-FMC, a lightweight federated content popularity prediction method based on Temporal Convolutional Networks (TCN) is introduced to collaboratively learn global content popularity while reducing prediction energy cost. The energy-aware utilities of the three involved parties are jointly formulated as a coupled non-linear optimization problem. To address this challenge, a two-stage game-theoretic algorithm is designed. Experimental results on a real-world testbed show that GT-FMC achieves up to 77.9% of Oracle in cache hit rate and 10.6%–32.4% reduction in transmission energy consumption compared to baseline methods. Complementary evaluations also validate the game-theoretic design’s effectiveness.