In the era of foundation models, Out-of-Distribution (OOD) problems, i.e., the data discrepancy between the training environments and testing environments, hinder AI generalization. Further, relational data like graphs disobeying the Independent and Identically Distributed (IID) condition makes the problem more challenging, especially much harder when it is associated with time. Motivated by this, to realize the robust invariant learning over temporal graphs, we want to investigate what components in temporal graphs are most invariant and representative with respect to labels. With the Information Bottleneck (IB) method, we propose an error-bounded Invariant Link Selector that can distinguish invariant components and variant components during the training process to make the deep learning model generalizable for different testing scenarios. Besides deriving a series of rigorous generalizable optimization functions, we also equip the training with task-specific loss functions, e.g., temporal link prediction, to make pre-trained models solve real-world application tasks like citation recommendation and merchandise recommendation, as demonstrated in our experiments with state-of-the-art (SOTA) methods. Our code is available at https://github.com/kthrn22/OOD-Linker