Does the principle of increased entropy support or challenge the Cosmological
Good question, and undoubtedly others are better equipped to give better or deeper answers. But I'll take a quick stab. First, there are at least several different categories of Cosmological Arguments, but I'm guessing you have in mind those involving design of some sort -- increasing entropy seems to suggest the cosmos tends toward disorder, which seems to undermine the notion that there is any sort of (intrinsic) or ultimate ordering. But now, with respect to design, what's to refute the idea that the increasing disorder is PART of the design, part of its aim? (the general problem with design arguments is that no one is ever very clear on just what the purported point of God creating the universe is .... But presumably introducing human beings to contemplate the universe (and God) is part of it, and why couldn't a cosmos with increasing entropy be something good for human beings to contemplate?) ... As far as other forms of Cosmological Args go, by my understanding they tend to involve intelligibility concerns: we couldn't make sense of motion, or of causation, or of the existence of the cosmos at all, unless there were a First Mover, Cause, or Necessarily Existent Creator. As far as I can see those forms of argument are neutral on the existence of entropy. So, in short, I can see where your intuition comes in -- but seems to me a lot more work has to be done before the fact of entropy would really raise a challenge for Cosmological Arguments.
hope that's a useful start!