The autonomous mobility-on-demand (AMoD) system plays an important role in the urban transportation system. The charging behavior of AMoD fleet becomes a critical link between charging system and transportation system. In this paper, we investigate a strategic charging pricing scheme for charging station operators (CSOs) based on a non-cooperative Stackelberg game framework. The Stackelberg equilibrium investigates the pricing competition among multiple CSOs, and explores the nexus between the CSOs and AMoD operator. In the proposed framework, the responsive behavior of AMoD operator (order-serving, repositioning, and charging) is formulated as a multi-commodity network flow model to solve an energy-aware traffic flow problem. Meanwhile, a soft actor-critic based multi-agent deep reinforcement learning algorithm is developed to solve the proposed equilibrium framework while considering privacy-conservation constraints among CSOs. A numerical case study with city-scale real-world data is used to validate the effectiveness of the proposed framework.