'maintain your cool' is a correct and usable phrase in written English.
You can use it when you're trying to encourage someone to remain calm in a difficult or stressful situation. For example, "I know it's hard, but try to maintain your cool and focus on finding a solution."
If you then add things like heat and humidity, then you have an added stress on the body by trying to maintain your cool body temperature.
When he got off the phone, he said to me: "It's hard to maintain your cool dealin' with three or four P.R. people.
The Blob (1958) may be a cult film but it doesn't show off McQueen to advantage – it's hard to maintain your cool when your co-star is a mass of flesh-eating gunge.
Shut your windows and doors during the day to keep hot air outside and maintain your cool indoor climate.
After having worked at a museum for two years, I know the feeling of struggling to maintain your cool while talking to museum goers about art you might not really enjoy.
Growing up in Los Angeles, you let famous folks scoot by all the time, with nothing more than an "I'm almost like you" glance, trying to maintain your cool.
Deal with difficult questions through humour, maintaining your cool, and sidelining ("come and talk to me after the presentation"), bouncing the question back to the questioner.
No matter if you're a size 5, a size 8, or a massive size 15, getting that protection on in the heat of the moment while maintaining your cool is key.
The key to maintaining your cool in this larger-than-life metropolis is to become neighborhood-focused. Craving great shopping and dining?
Many times work place conflict is impossible to avoid; however, you can stop its escalation by maintaining your cool in the eye of the storm.
Although NBC's new sitcom is only halfway through its first season, it's already captured the cluelessness and frustrations of becoming thirty-something parents with nostalgic soul and jocular fun, especially zoning into that "new parent" struggle between accepting domestication and desperately maintaining your cool self-identity.
