
Kevin C. answered 04/05/19
Theology Geek: Greek, Latin, Proofreading, Writing, ACT-SAT-GRE
Thanks for asking that question. It's a very good one. The fallacy of "special pleading" involves ignoring unfavorable aspects of the argument. Perhaps some theists do this. But Thomas Aquinas does not ignore unfavorable aspects of any argument. In fact, in his summae, he first lists the arguments against his position as he knows them.
The cosmological argument for the existence of God is essentially Thomas Aquinas's fifth way of proving God's existence. Thomas's articulation systematically presents the evidence better than most contemporary theists, so I'll offer what he says. He writes,
The fifth way is taken from the governance of the world. We see that things which lack intelligence, such as natural bodies, act for an end, and this is evident from their acting always, or nearly always, in the same way, so as to obtain the best result. Hence it is plain that not fortuitously, but designedly, do they achieve their end. Now whatever lacks intelligence cannot move towards an end, unless it be directed by some being endowed with knowledge and intelligence; as the arrow is shot to its mark by the archer. Therefore some intelligent being exists by whom all natural things are directed to their end; and this being we call God.
Summa Theologiae, I, q. 2, a. 3, resp. (http://www.newadvent.org/summa/1002.htm#article3)
First I will explain this and then I will address your question why there is no fallacy of special pleading.
The notion around which this proof turns is what we might call "teleological ordering." Things that are not intelligent move with direction toward some telos (goal, end). This is not the observation of some interior beauty in the cosmos, but the governance of that cosmos with intention and will. And this is not merely with respect to individual examples, such as the bombardier beetle or the venus fly trap, but all things in the so-called lower creation act with intention and ordering beyond their individual, specific capacities as though directed by an orderer or governor.
Now, why does this not involve some sort of special pleading? If God himself needed an orderer, then he would not be God. That thing ordering or governing him would be God. And this is why Christian metaphysics has held that God's essence and existence to be one and the same. Now your question also touches upon causality, which Aquinas takes up in his second proof of divine existence (efficient causality: http://www.newadvent.org/summa/1002.htm#article3). But the notion is one and the same: God does not need a cause. If God were to need a cause, he would not be God. The cause of god would then be the true God.
I hope that helps with your question.