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This paper studies the problem of information freshness-aware task offloading in an air-ground integrated multi-access edge computing system, which is deployed by an infrastructure provider (InP). A third-party real-time application service provider provides computing services to the subscribed mobile users (MUs) with the limited communication and computation resources from the InP based on a long-term business agreement. Due to the dynamic characteristics, the interactions among the MUs are modelled by a non-cooperative stochastic game, in which the control policies are coupled and each MU aims to selfishly maximize its own expected long-term payoff. To address the Nash equilibrium solutions, we propose that each MU behaves in accordance with the local system states and conjectures, based on which the stochastic game is transformed into a single-agent Markov decision process. Moreover, we derive a novel online deep reinforcement learning (RL) scheme that adopts two separate double deep Q-networks for each MU to approximate the Q-factor and the post-decision Q-factor. Using the proposed deep RL scheme, each MU in the system is able to make decisions without a priori statistical knowledge of dynamics. Numerical experiments examine the potentials of the proposed scheme in balancing the age of information and the energy consumption.