Can a father leave the state of Florida to the state of Arazona
Without telling his mother

Is there an order awarding custody?

without telling his mother

I assume you meant the mother of his child. Not sure why you would care whether he told HIS mother.

If the father the non custodial parent and he is not taking the child, then he is under no compulsion to keep the mother informed of his whereabouts. However, if there is a child support order, then he needs to continue paying support.

If you want a better answer you need to explain the while situation.