I recently purchased and installed a Honeywell WP6500 WiFi thermostat and thanks to Stack Exchange, got it up and running. I have a house in New Hampshire, which I use occasionally during the winter. I do not like to keep heat on in the house all winter as I may not be there for 2 to 3 weeks. When I set the thermostat to maintain the house at 41°F it works fine. However, if I set it to the heat off position, at some point it stops communicating with the router, and I cannot turn the system on or raise the temperature. It can get 5 to 10°F in the house. I have tried on many occasions to get some answers from Honeywell as to, is there a temperature at which the thermostat, stops communicating with the router, and if there is, can it be overridden.
You're looking for a specification called the operating temperature range of the thermostat or router or both. This is the temperature range within which the manufacturer states that the device will function correctly.
For consumer electronics like these, it's typically from 0°C (32°F) up to about 60°C (140°F) to 70°C (158°F). So if you turn the heat to off, the ambient temperature in the house will drop below that range, and the thermostat will stop working properly.
There should be a specifications page in the manuals for the thermostat and router that will tell you their minimum operating temperatures. Keep your house above the upper of those temperatures and you should be OK.
I highly doubt that Honeywell would design the system on purpose to disconnect from the network based on the ambient temperature (BUT: I may be wrong about this!).
That said, it may be that either your router or the thermostat are starting to have technical difficulties if the temperature drops too low. If you have one lying around, try a different router, or maybe it's enough to just change the channel setting on the one you have. Or move it closer to the thermostat, even if it's just for testing by running a long cable through the house.
Some background: Both the router and the thermostat need to have a chip inside (called clock source) that gives them a reference for generating the ~2.4 GHz signal to communicate. Kind-of like a metronome that a piano player uses to play at the right bpm. The frequency of the reference signal that these chips generate is dependent on the temperature of the chip, which in turn depends on the ambient temperature in your house. Depending on how cheap the chip is, this change in reference might be enough for the two devices to no longer be able to talk to each other reliably. Think of slowly turning the dial on your radio off the station that you are listening to. You might still hear music for a little while, but eventually it will cut out.
If you find out it's the router, one thing you can do is stick it in a cardboard box. That will trap the heat that the router generates inside and keep it a little warmer. Just make sure that it doesn't overheat when it's actually warm in the house. Unfortunately, if it turns out to be the thermostat's fault, you can't stick that in a box, since then it won't be able to measure the outside temperature any more. But swapping out your router (for a cheaper one) might still fix it. At what temperature the system stops working depends on the exact build of the involved devices.