No they'll survive just fine without, sloshing like evolutionary sludge into every nook and cranny that supports life as they always have.

We don't need tech to survive either, so long as we don't mind living like animals!

Or they won't. Sixth mass extinction.

I'm pretty sure animals have much bigger issues to worry about than outside temperature, like loss of habitat.

Especially the Pika, which needs very specific temperatures that are only found at high elevations to maintain thermoregulation. Global warming means that they will literally die of heat exhaustion during future summers as the temperature increases at higher elevations.

https://en.wikipedia.org/wiki/Pika

Edit: specifically the American Pika

https://en.wikipedia.org/wiki/American_pika