He 'kept/maintained his cool' [=he didn't get angry or upset] even though it was clear that he was being unfairly treated.
He was struggling to 'maintain his cool/composure'. [=he was struggling not to become angry or upset]
They barely had enough food to 'maintain life'. [=to make it possible for life to exist or continue]
(..) ...and finds your wife in bed with her lover... ...riddled with .-caliber bullets. Does that strike you as a fantastic coincidence, or is it just me? Yes, it does. Yet you still maintain you threw your gun into the river... ...before the murders took place. That's very convenient. It's the truth. (..)1994 The Shawshank Redemption