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Cooperative coordination at unsignalized road intersections, which aims to improve the driving safety and traffic throughput for connected and automated vehicles, has attracted increasing interests in recent years. However, most existing investigations either suffer from computational complexity or cannot harness the full potential of the road infrastructure. To this end, we first present a dedicated intersection coordination framework, where the involved vehicles hand over their control authorities and follow instructions from a centralized coordinator. Then a unified cooperative trajectory optimization problem will be formulated to maximize the traffic throughput while ensuring the driving safety and long-term stability of the coordination system. To address the key computational challenges in the real-world deployment, we reformulate this non-convex sequential decision problem into a model-free Markov Decision Process (MDP) and tackle it by devising a Twin Delayed Deep Deterministic Policy Gradient (TD3)-based strategy in the deep reinforcement learning (DRL) framework. Simulation and practical experiments show that the proposed strategy could achieve near-optimal performance in sub-static coordination scenarios and significantly improve the traffic throughput in the realistic continuous traffic flow. The most remarkable advantage is that our strategy could reduce the time complexity of computation to milliseconds, and is shown scalable when the road lanes increase.