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.
You can change a setting called BSS-minrate under WLAN advanced settings.
Set the value to 12Mbps.
Setting BSS-minrate will shrink the AP-Client management coverage area.
This means, if a client cannot sent the management traffic to AP with at least 12Mbps speed, it will roam to another AP. This way client will mostly connect to an AP where it can send and receive management traffic with at least 12Mbps with better signal strength (closest AP).
This will help remediate sticky client behavior.
Also AP placement, AP TX power and a site survey is the key for optimum performance on a WiFi network.