Trusted open set recognition aims to classify known classes and reject unknown ones, as well as outputs an uncertainty estimate to measure the reliability of recognition results, thus extending the application scenarios of traditional open set recognition methods to risk-sensitive fields. Current methods assume that the covariate distribution of the known classes remains constant during training and testing. However, due to the common occurrence of covariate shift in practical applications, existing methods often suffer from limited generalization. To this end, a causal evidence learning framework, highlighted by the controllable Evidential Uncertainty Guided Adversarial Data Augmentation (EUG-ADA) and Causal Adversarial Disentanglement (CausalAD) strategies, is proposed to support trusted open set recognition under covariate shift. Specifically, EUG-ADA generates high-quality augmentation samples to increase training data diversity, guided by controllable evidential uncertainty and constrained by semantic consistency. Moreover, it is complemented by the CausalAD, which learns causal representations through causal intervention, mitigating the risk of misrecognition of unknown classes caused by the model's reliance on shortcuts for prediction. The combined effect of EUG-ADA and CausalAD enables the model to learn more generalized and robust causal evidence for trusted open set recognition. Finally, extensive experimental results on both real-world and synthetic data validate the effectiveness of the proposed method, demonstrating that it improves not only open set recognition performance under covariate shift but also the reliability of uncertainty estimates. The code is released on https://github.com/ScorpioBao/CEL-OSR.