First, given how bitter their rivalry was and how much a threat they posed to each other, I doubt either Carthage or Rome would have gone for anything but an absolute, devastating victory in the Second Punic War.
If we suppose a somewhat different situation, where Rome and Carthage fight each other to a standstill, both become too exhausted to fight on, and have to stop killing each other long enough to recover, then a "cold war" might be a possibility. I can't imagine that they would carry it out with the indirectness that a "cold war" implies, however. They would probably wage direct war with each other, possibly over territories like Greece or Egypt, but I think that one would end up destroying the other eventually. Eventually, one of them is bound to get wrapped up in a civil war or some other crisis, and the other will probably capitalize on the opportunity to cripple their enemy. The Mediterranean just wasn't big enough for the two of them by the Second Punic War. An ancient Mediterranean with Carthage and Rome competing for longer would be interesting, but probably would end before too long and couldn't be called a "cold war."