What type of smart theromstat allow local only operation? The popular ones I've looked at (nest, ecobee, honeywell) all look like they require internet connection to work.
Look at this list and go into each page to see the “IoT class”.
“Local push” is the best but “Local polling” would satisfy your requirements as well.
Either way, HA lets you use a whole load of random ZWave/ZigBee/whatever thermostats:
https://community.home-assistant.io/t/latest-and-greatest-th...