Necessarily yes, if it applies here and now, it will appear everywhere at every time. Metaphysics aren´t location-bound, but an all-or-nothing explanation of the realtionship of the different grounds of reality and their relationship. This is a horrible mouthful, so to explain it in more understandable forms, metaphysics are the laws that underlie reality and thus are even more fundamental than the laws of physics that we know of. So if the principle of causality which you quoted applies here, it necessarily has to apply at every corner of the universe, if it doesn´t apply there, it doesn´t apply here. This will also lead to the conclusion, that if we discovered a hypothetical part of the universe which only contained unknown particles and none of those known to us, we would conclude that this part also underlies the logical principles of change and causality, because “brute facts” with no explanation in principle for their existence and/or attributes are unknown to us and logically impossible and incoherent, if we believe, and I´d argue that we would have every reason to, given our scientific and technological progress, as well as our ability to predict particles with mathematical models, which are verified later (the best example would be the Higgs-Boson), that we discover real and objective facts about the universe. Like I mentioned, we would come to that presuppositionn intuitively because of our experiences thus far, but also logically, because one of the metaphysical presuppositions in science is exactly the causality principle you quoted.
For the sake of it let´s imagine a universe where the principle doesn´t apply. The brute fact would make it impossible to let arise a structure which has more than the mere appearance of rationality, because, we have seen, that the brute fact has no explanation for any of its attributes, which would lead to no explanation of the attributes of other things. In fact, we would have to expect that there would be no uniformity of the attributes of the same things (e.g. the mass of protons, the electrical charge of an electron). Why? Because the things always exist in reference to the brute fact and are even relying on it, but that would also mean that there would be no possible explanation for their appearance in principle, which also includes these attributes being constants. Either way, what we would have discovered so far and will in the future would be mere appearances of rationality, but unable to give us any fundamental facts about the universe. Another fact: If the universe´s structure isn´t rational, that would include our brain, which would make us unable to get any more facts, since we can´t ever be sure if the conclusion we have reached could ever be right, because the methods aren´t trusthworthy since they are used or have been set up by an irrational mind. The circularity of that approach should be obvious, cognitive skepticism stops any progress and I would argue that this exact progress which obviously enabled us to alter our environment for the better (technology) or worse (pollution and trash), which has been built upon a mathematics which presupposed its own rationality, is a good enough refutation of the approach of cognitive skeptics and proponents of brute facts. Once again a phrase I love applies: Their position gives birth to the skepticism of their own position.These can´t be reached by any possible logical or rational analysis, but if they would be right, we would only know by luck.