It's a cold war, I'm a colder soldier
i.e. he is tough and strong enough to survive through all his problems.
A cold war is a state of political tension that does not involve the guns, armies, battlefields, etc, of a “typical” war. Common is saying he’s living in a state of tension in a rough life.
If you want to survive you have to be stronger than your enemy (being colder than cold).
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