This can't be an issue with any cache on AP, decision to which AP to connect is done by client.
Some clients may remember AP MAC, and to reconnect to other AP need to be disconnected and connected to network. It may apply to thermostat, as it is not expected to move around and may have not implemented things related to roaming. I have seen it with wifi print-servers and payment card terminals -- they stayed on remote AP when moved to different location, until restarted and connected to network again. As you never restarted thermostat, it can be a reason.
If it happens with different clients too (such as mobile phones and laptops), than investigate channels used by APs, signal levels, interference, modulation used, etc.