An objection I hear a lot to the KCA is that it commits the fallacy of special pleading because it exempts God from the same causal principle that it applies to the universe. If you say the universe requires a cause but you exempt God from requiring a cause, then you're committing the fallacy of special pleading.
This is not a sound criticism. The KCA reaches the conclusion that the universe has a cause from the premise that the universe began to exist, and there are various arguments meant to demonstrate that the universe began to exist. Unless it can be shown that God began to exist, then one cannot draw the same conclusion about God. The arguments that apply to the universe do not apply equally to God, so God is not being arbitrarily exempted from requiring a cause.
The special pleading fallacy can only be invoked when a person makes an arbitrary exception to a general principle (usually solely to avoid an unwanted conclusion). Suppose the first premise in the KCA was, "Everything requires a cause." Well, then it might be special pleading to exempt God, but that's not what the first premise to the KCA says. Rather, it says, "Everything that begins to exist requires a cause." That's why the second premise attempts to show that the universe began to exist. Unless the argument can show that the universe began to exist, it cannot draw the conclusion that the universe requires a cause. It isn't enough for the universe to exist in order to draw that conclusion; it must exist and have a beginning.
But then somebody will say the first premise itself commits the fallacy of special pleading merely by the use of the qualification, "that begins to exist." But this is not an arbitrarily qualification. There is a justification for it. A thing cannot be brought into existence if it already exists. It can only be brought into existence if it doesn't already exist. So if the universe had a beginning and God did not, then there'd be a reason to think the universe needed a cause but God did not. Since there is a justification for the exception or qualification, it doesn't commit the fallacy of special pleading.
Of course there are cosmological arguments that say even things that always exist may require causes. Some people believe God is continuously causing the universe to exist. These people believe that if God removed his sustaining power of causation, the universe would vanish. The first premise in the KCA doesn't deny this. It just makes a more modest claim. At least those things which come into existence require a cause. The first premise doesn't say anything about those things that do not begin to exist. If a person wanted to accuse a cosmological argument of special pleading, they might ought to attack those other arguments, not the KCA.
But even in the case of those other arguments, the expanded causal principle doesn't arbitrarily exclude God. Instead, they make a distinction between necessary things and contingent things. Contingent things require causes and necessary things don't. And again, this distinction is not arbitrarily. The reason necessary things do not require causes is because it's impossible for them to not exist. That's what it means to be necessary. Whatever reasons there are to think contingent things require causes wouldn't apply to necessary things. One reason, for example, might be that it is possible for contingent things to not exist, and that obviously doesn't apply to necessary things.
I think the primary reason the accusation of special pleading comes up so much is because in an attempt to dismiss all cosmological arguments in one go, people kind of lazily lump them all together in some distorted representation that actually does commit the fallacy of special pleading. They'll say something like, "The basic cosmological argument is this: Everything requires a cause; therefore the universe requires a cause; so God exists; but God doesn't have a cause." I got this same impression in my freshmen philosophy class when I first heard "the uncaused cause" argument that was based on Aristotle. But according to Ed Feser, no prominent philosopher in the history of philosophy has ever made that argument, not even Aristotle ("So you think you understand the cosmological argument?" and "Straw men and terra-cotta armies"). It's a straw man, and I think a lot of people want to shoe horn the KCA in such a way that the special pleading fallacy applies because it's an easy way to dismiss the argument without having to address the premises or the defenses of the premises. But just a little reflection ought to make it obvious that the criticism doesn't apply.
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