Here it is used sarcastically .He's like Angelina Jolie with a pet cause.= he keeps changing his interests/projects all the time

Could you tell us the context it was used in? I ask because it could also be used to describe a person’s dedication to something (after all, Angelina Jolie is a bleeding heart celebrity and she really is dedicated to most of her pet causes). Like azimux said, the “cause” refers to a charitable goal or project (that someone is dedicated to) and “pet” is used to mean something that is valued very much (in this case, more than other causes). For example, a “pet project” is an activity someone pursues as their personal favorite. So a “pet cause” would be a charitable goal that someone supports more deeply than the other causes they are interested in. The sentence “he's like Angelina Jolie with a pet cause” could mean that the “he” is really dedicated to whatever he’s doing. But depending on the context, it could also be a sarcastic comment.