Considering the enormous potential application of autonomous mobility-on-demand (AMoD) systems in future urban transportation, the charging behavior of AMoD fleets, as a key link connecting the power system and the transportation system, needs to be guided by a reasonable charging demand management method. This paper uses game theory to investigate charging pricing methods for the AMoD fleets. Firstly, an AMoD fleet scheduling model with appropriate scale and mathematical complexity is established to describe the spatio-temporal action patterns of the AMoD fleet. Subsequently, using Stackelberg game and Nash bargaining, two game frameworks, i.e., non-cooperative and cooperative, are designed for the charging station operator (CSO) and the AMoD fleet. Then, the interaction trends between the two entities and the mechanism of charging price formation are discussed, along with an analysis of the game implications for breaking the non-cooperative dilemma and moving towards cooperation. Finally, numerical experiments based on real-world city-scale data are provided to validate the designed game frameworks. The results show that the spatio-temporal distribution of charging prices can be captured and utilized by the AMoD fleet. The CSO can then use this action pattern to determine charging prices to optimize the profit. Based on this, negotiated bargaining improves the overall benefits for stakeholders in urban transportation.