That's an excellent question! It's a difficult question to answer, because different animals almost certainly respond differently to such situations,and because it's very difficult to design experiments that test such questions.

My impression is that some animals probably do reduce their activity when they detect a storm coming, although it's not clear whether they're actually detecting pressure changes or some other cues. However, other animals are most likely clueless about pressure changes or to approaching storms, while other animals probably get more active. For example, I can imagine animals with very fast metabolisms (so they need to eat very often), scrambling to find and eat as much food as possible before being forced to take shelter for an unknown period of time. Also, if some animals slow down and stay put,there are probably other predatory animals that take advantage of that and increase their hunting in order to find and catch those animals.

I speculate on all these things, because there's not a lot of research that focuses on these questions. Asking whether or not an animal can detect pressure changes requires a lot of laboratory research. Even if it's determined that they can detect pressure changes, that doesn't prove that the animal uses that information when a storm approaches. I don't know how you would answer that question with wild animals--if you try to answer it with caged animals, there's no guarantee that captivity didn't change the animal's behavior. So it's a very difficult puzzle to solve!