What, according to Domat, gives kings "legitimate authority?"
I assume that you are talking about Jean Domat, a French thinker who wrote in the 1600s. Domat was a propoenent of what we now call the "divine right of kings." As you can see from this phrase, Domat thought that the "legitimate authority" of a monarch was given by God.
Domat believed that the citizens of a state needed to be absolutely obedient to their rulers in the same way that people needed to be absolutely obedient to God. He argued that God had instituted government among people and that to disobey the sovereign was the same as to disobey God.
Domat says, for example,
the power of princes comes to them from God and is placed in their hands as an instrument of his providence and his guidance of the states...
He also says that
Since government is necessary for the common good and God himself established it, it follows that those who are its subjects must be submissive and obedient. For otherwise they would resist God
From these passages, it is clear that Domat thinks that kings get their authority from God.