Yes but no: The original device operates in the 868MHz band, which is a ISM band in the EU. However this band isn't availabe for public use in the US (IIRC some mobile phone cells operate in that band) - so the original thermostat is already illegal to sell/buy/use in the US.

If you bought such a thermostat in the US (e.g. at walmart or what-do-I-know), that device would probably operate on 433 or 915 MHz. I think you can legally broadcast on these in the US, much like the OP did [in the UK]. Power limits might be a bit different and such. But IANAL and no US person, so do some research before grabbing a HackRF One ;-)