Short-term customer load forecasting is vital for the normal operation of power systems. Unfortunately, conventional machine learning-based forecasting methods are susceptible to generalization issues (e.g., the customer heterogeneity and distribution drift of load data), manifested in model performance degradation. In recent years, some studies have employed the advanced deep learning technology, such as online learning, to overcome the aforesaid problems. However, these methods can only alleviate the adverse impacts of generalization problems on model performance, because they are inherently built on unstable relationships (i.e., correlations). In this paper, we propose a novel causal inference-based method to improve the generalization for short-term customer load forecasting models. Specifically, we first investigate the causal relations between input features and the output in existing methods, and introduce the load characteristics as an extra model input to enhance the causality. Then, we closely inspect the causality in models by using the causal graph to distinguish the confounder, followed by employing the causal intervention with do-calculus to eliminate the spurious correlations caused by the confounder. Moreover, we propose a novel load forecasting framework with the load characteristic extraction, characteristic pool approximation and characteristic-injected model to realize the causal intervention in an efficient and fidelity way. Finally, the effectiveness and superiority of our proposed method are validated on a public dataset.