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This work studies a real-time environment monitoring scenario in the industrial Internet of things, where wireless sensors proactively collect environmental data and transmit it to the controller. We adopt the notion of risk-sensitivity in financial mathematics as the objective to jointly minimize the mean, variance, and other higher-order statistics of the network energy consumption subject to the constraints on the age of information (AoI) threshold violation probability and the AoI exceedances over a pre-defined threshold. We characterize the extreme AoI staleness using results in extreme value theory and propose a distributed power allocation approach by weaving in together principles of Lyapunov optimization and federated learning (FL). Simulation results demonstrate that the proposed FL-based distributed solution is on par with the centralized baseline while consuming 28.50% less system energy and outperforms the other baselines.