While Augustine (and Cyprian) believed in predestination, I doubt, either of them believed in "double predestination". I see nothing in their works supporting that view. I noticed the following on the Catholic website (NewAdvent) about
predestination:
Predestination (Latin præ, destinare), taken in its widest meaning, is every Divine decree by which God, owing to His infallible prescience of the future, has appointed and ordained from eternity all events occurring in time, especially those which directly proceed from, or at least are influenced by, man's free will.
With the exception of "has appointed and ordained from eternity all events" it is very similar to what CB was saying.
And, as always, I'm happy to promote the church fathers since they rightfully believed in the inerrant scriptures as the only source of infallible writings.