Anomaly detection (AD) is the machine learning task of identifying highly discrepant abnormal samples by solely relying on the consistency of the normal training samples. Under the constraints of a distribution shift, the assumption that training samples and test samples are drawn from the same distribution breaks down. In this work, by leveraging tools from causal inference we attempt to increase the resilience of anomaly detection models to different kinds of distribution shifts. We begin by elucidating a simple yet necessary statistical property that ensures invariant representations, which is critical for robust AD under both domain and covariate shifts. From this property, we derive a regularization term which, when minimized, leads to partial distribution invariance across environments. Through extensive experimental evaluation on both synthetic and real-world tasks, covering a range of six different AD methods, we demonstrated significant improvements in out-of-distribution performance. Under both covariate and domain shift, models regularized with our proposed term showed marked increased robustness. Code is available at: https://github.com/JoaoCarv/invariant-anomaly-detection