Depends on which species you have and just how cool the cooler greenhouse will be.

Archontophoenix alexandrae, and at night it gets down to around 15C but during the day it's about 25-30C

The greenhouse should be okay. Keep the humidity up. Alex is a rainforest plant so likes the humidity without extremes of temperature. If you get some of the savanna/woodland species, like a lot of the Livistonas, they'd really appreciate the 35's right through and not be worried by lower humidity.

Thanks for the advice. I am a Huge fan of Alex's, Bangalows and Foxtail type palms.
i love the look but the only issue is where i am Bangalows are the only readily available palms.

My sister lives in Melbourne and I sent her a Foxtail seedling from one of my trees. Suggested she keep it indoors in colder weather but it never survived. But of course, she's not a plant person so I shouldn't have expected anything. You certainly don't see many Foxtails around there anyhow.