Robotic policy learning for complex real-world manipulation tasks has seen rapid recent progress, enabled in large part by the ability to collect demonstrations through human operation. However, policies trained from such demonstrations often execute tasks far more slowly than the robot's physical capabilities, as demonstration data is collected under practical constraints that favor conservative, success-oriented trajectories over execution speed. Existing policy acceleration methods determine execution tempo through data preprocessing or heuristic rules, rather than learning execution speed optimized for the task. In this paper, we propose SpeedAug, a policy acceleration framework that enables policies to learn task-optimal execution tempo via reinforcement learning (RL). SpeedAug first learns a tempo-enriched prior policy from speed-augmented demonstrations that captures diverse execution tempos. Building on this tempo-enriched prior, RL fine-tuning guides exploration to refine action trajectories and optimize execution tempo efficiently. Experiments on robotic manipulation benchmarks demonstrate that SpeedAug substantially improves the sample efficiency of policy acceleration while maintaining high success rates, achieving fast and stable task execution. Applied to a real-world manipulation task, SpeedAug improves task throughput by 1.8x using only 16 minutes of online interactions without compromising the success rate.