Good luck with the experiments. Depending on the situation, you might be able to replace your thermostat with one that is controllable by local automation systems and doesn't require cloud access. I'm not personally familiar with the various products on the market, so I can't recommend anything (and besides, your HVAC system itself may or may not present compatibility issues/questions).
Let us know what you find once you've got some results from your testing.
And if/when your problem is solved, please consider marking this topic as [Solved]. See How to mark a topic as [Solved] for a short how-to.
(in truth, your problem may never be 'solved' but you can mark it as such and include an explanation of your findings -- still could be useful for future readers)