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]
(..) or you're just a vigilante. - Is Fox still here? - Yes, sir. We need to send these people away. Those are Bruce Wayne's guests. You have a name to maintain. I don't care about my name. It's not just your name, sir. It's your father's name. And it's all that's left of him. Don't destroy it. (..)2005 Batman Begins