I tend to leave the router on, as on the few times that I have switched it off my line speed has been reduced to 2.0mps and no amount of use will get it increased via Dynamic Line Management. I then have to speak to the ISP confirm that I have done all of the checks and they then contact BT who switch the speed up and from thereon it is fine. However recently we went to India and prior to leaving we had been getting intermittent power outages from the main RCD tripping. Despite numerous test our electrician could not find any fault in the circuits. So as we wanted to keep the heating on continuously we disconnected every appliance except the boiler, HD recorder and answerphone. Our instructions to the kids were that if the temperature dropped below zero to ring the house phone. If the answerphone responded all was okay, if it didn't then to drive over and check/reset the RCD.
This seemed a simpler and more reliable approach than sitting a webcam pointing at the boiler instrument panel, or trying to use a web app in deepest India. Both of which required 240v supplies from somewhere.
On return the router came straight up to its normal line speed of 5-6mps and has been stable for the last week since we got back.