Self-adaptive robots (SARs) in complex, uncertain environments must proactively detect and address abnormal behaviors, including out-of-distribution (OOD) cases. To this end, digital twins offer a valuable solution for OOD detection. Thus, we present a digital twin-based approach for OOD detection (ODiSAR) in SARs. ODiSAR uses a Transformer-based digital twin to forecast SAR states and employs reconstruction error and Monte Carlo dropout for uncertainty quantification. By combining reconstruction error with predictive variance, the digital twin effectively detects OOD behaviors, even in previously unseen conditions. The digital twin also includes an explainability layer that links potential OOD to specific SAR states, offering insights for self-adaptation. We evaluated ODiSAR by creating digital twins of two industrial robots: one navigating an office environment, and another performing maritime ship navigation. In both cases, ODiSAR forecasts SAR behaviors (i.e., robot trajectories and vessel motion) and proactively detects OOD events. Our results showed that ODiSAR achieved high detection performance—up to 98% AUROC, 96% TNR@TPR95, and 95% F1-score—while providing interpretable insights to support self-adaptation.