Foundation models for computational pathology are expected to facilitate the development of high-performing, generalisable deep learning systems. However, in addition to biologically relevant features, current foundation models also capture pre-analytic and scanner-specific variation that bias the predictions made by downstream task-specific models trained on these features. Here we show that introducing novel robustness losses during downstream model training reduces sensitivity to technical variability. A purpose-designed comprehensive experimentation setup with 27,042 whole-slide images from 6,155 patients is used to train thousands of models from the features of eight well-known foundation models for computational pathology. In addition to a substantial improvement in robustness, our approach improves classification accuracy by focusing on biologically relevant features. It mitigates robustness limitations of foundation models for computational pathology without retraining the foundation models themselves, enabling development of models that are more suitable in real-world clinical use.