Is this that Honeywell REDLink stuff? When I got my heat pump installed, they put that crap in. It was ill behaved on my network. It needs the Internet so... you can manage the thermostats from the Honeywell all on your phone? I think?

You have a lot of options for thermostats. I think it does want some kind of external temperature to know when it's too cold to run the heat pump. I ended up re-installing the Ecobee thermostats that I used to have; these are probably not any better from a privacy perspective. But I had a long conversation with my installer, and he distrusted them because they got their external temperature from "the internet", which he felt was risky. "What if your network goes down? What if the temperature at your house is different than at the airport, or wherever they measure the temperature?"

In reality, the external thermostat was also risky the way they set it up, because "what if the sun is shining directly on it and you get an inaccurate reading?"

No idea about Honeywell stuff, but last time I was stuck with something that needed wifi to work, I bought a cheap used router, made an access point just for that device, and never connected the router to the internet.

You may be able to run a wire out to an outdoor thermometer so your thermostat can just read the temperature outside.