Large Language Models (LLMs) encapsulate a surprising amount of factual world knowledge. However, their performance on temporal questions and historical knowledge is limited because they often cannot understand temporal scope and orientation or neglect the temporal aspect altogether. In this study, we aim to measure precisely how robust LLMs are for question answering based on their ability to process temporal information and perform tasks requiring temporal reasoning and temporal factual knowledge. Specifically, we design eight time-sensitive robustness tests for factual information to check the sensitivity of six popular LLMs in the zero-shot setting. Overall, we find LLMs lacking temporal robustness, especially to temporal reformulations and the use of different granularities of temporal references. We show how a selection of these eight tests can be used automatically to judge a model's temporal robustness for user questions on the fly. Finally, we apply the findings of this study to improve the temporal QA performance by up to 55 percent.