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Industrial wireless networks (IWNs) have attracted significant attention for providing time-critical delivery services, which can benefit from device-to-device (D2D) communication for low transmission delay. In this article, a distributed scheduling problem is investigated for D2D-enabled IWNs, where D2D links have various age-of-information (AoI) constraints for information freshness. This problem is formulated as a constrained optimization problem to optimize D2D packet delivery over limited spectrum resources, which is intractable since D2D users have no prior knowledge of the operating environment. To tackle this problem, in this article, an AoI-aware scheduling scheme is proposed based on primal-dual optimization and actor--critic reinforcement learning. In specific, multiple local actors for D2D devices learn AoI-aware scheduling policies to make on-site decisions with their stochastic AoI constraints addressed in the dual domain. An edge-based critic estimates the performance of all actors' decision-making policies from a global view, which can effectively address the nonstationary environment caused by concurrent learning of multiple local actors. Theoretical analysis on the convergence of learning is provided and simulation results demonstrate the effectiveness of the proposed scheme.