Machine learning enables powerful cosmological inference but typically requires many high-fidelity simulations covering many cosmological models. Transfer learning offers a way to reduce the simulation cost by reusing knowledge across models. We show that pre-training on the standard model of cosmology, ΛCDM, and fine-tuning on various beyond-ΛCDM scenarios — including massive neutrinos, modified gravity, and primordial non-Gaussianities — can enable inference with significantly fewer beyond-ΛCDM simulations. However, we also show that negative transfer can occur when strong physical degeneracies exist between ΛCDM and beyond-ΛCDM parameters. We consider various transfer architectures, finding that including bottleneck structures provides the best performance. Our findings illustrate the opportunities and pitfalls of foundation-model approaches in physics: pre-training can accelerate inference, but may also hinder learning new physics.