Our kid went through that phase. He grew out of it.
I don't call it lying. I call it "adjusting the world to be what you want it to be." Which isn't realistic, but that's what it is. Divorce is traumatic, and there are two possibilities. 1. It was a lie. 2. It was easier to say it was a lie than push the matter and all the consequences that would have. Is the "truth" so important? Isn't the "truth" a matter of perspective?
If you're thinking, "But this is serious," yes, it is, but remember that there is going to come a time when he tells you something that is not a lie, and you refuse to believe him, and you'll wind up feeling terrible!
Unless he has a reall live "conduct disorder,." which has nothing to do with Asperger's, and requires a psychiatrist to make that determination, you should check everything, but don't automatically disbelieve him. Do it openly. When he realizes that you are going to check everything, first he's going to yell "you don't trust me." Answer, I don't trust anyone. If I check and what you say is true, I'll do something about it. If it isn't, you can expect to have a discussion with me."
It is not easy advice to take, and the "discussion" should not always include consequences for a lie. Amazing, what happens when a child knows you're checking on them and not taking their word for something. And by the way, it's also a characteristic of Neuro Typical children.
Isn't child rearing fun?
btdt