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Carbon trading market is a central policy tool for promoting cleaner enterprises' energy consumption and investment behaviors. It is crucial for enterprises to establish a green and low-carbon development model by clarifying carbon trading impact mechanism on enterprise behaviors and the interactive relationship between entities in the carbon market. China's contract-based carbon market is still in the development stage, the impact of the mainstream allowance mechanism and government regulation on enterprises' energy behaviors has yet to be determined. In view of this, this paper describes mathematical mechanisms of the influence of carbon prices and carbon quotas on enterprises' energy consumption and investment, and the evolutionary game model of government-enterprise behaviors is constructed, the evolution process and appropriate strategies of enterprises' energy behaviors in the development of the carbon market are discussed. This paper uses real data of China carbon market to simulate the model, it is found that strategies such as relaxation of carbon quotas and carbon subsidies dynamization are needed during the initial development of the market. In the mature stage of the carbon market, strategies are needed to increase the proportion of government carbon revenue and reduce its regulatory cost. The higher the initial willingness of government-enterprise, the better the effect of strategies and the constraints of public willingness. The model and results have theoretical implications for enterprises' energy transformation and government regulation, and provide a reference for the development of China's carbon market. • Constructed benefit-cost analytical framework for enterprises' energy behaviors. • Described mechanisms of carbon trading affecting enterprises' energy behaviors. • Constructed government-enterprise behaviors model by the evolutionary game theory. • Found new conclusions of enterprises' energy behaviors under the carbon market. [ABSTRACT FROM AUTHOR]