eHoefler wrote:
It cost less to maintain temperature than to raise it from cold.
On what do you base this claim?
For a normal water heater, it's exactly the opposite way around, and a simple thermodynamic analysis shows why. The rate of heat transfer between two bodies is proportional to the difference in their temperatures. By maintaining the temperature of the water in the heater at a high temperature, the temperature difference between the heater and the surrounding environment is kept at a maximum, and the heat that escapes from the heater is maximized. If it gets cooler, on the other hand, the rate of transfer goes down and the total heat loss goes down.
It should be pretty obvious that whatever heat is lost is what needs to be put back into the water by the heater, plus some factor for inefficiency in the heating system. For electric heating, the inefficiency is basically negligible; for gas, it's not negligible. (Gas would also be more efficient at heat transfer when the water is cool than when its warm, since the temperature difference between the flue and the water is greater, and then the efficiency drop a bit as the temperature of the water goes up.)
On a quite long term basis, it's very obvious that leaving the water heater on is less efficient than not. If you left it on all summer, say, you'd probably run out of propane in the RV tank; while, with it off, the propane level would not change.
For a heat pump system (such as some residential water heaters use), the analysis may get rather more complicated because the efficiency of the heat pump is not fixed and because most of them have a hybrid system where there's also an electric heating element. A big demand (like heating the tank up from cold) would involve the heating element and extended operating of the heat pump, which would get at least somewhat less efficient as the heat from the surrounding environment is moved to the water. The ambient temperature around the heater goes down, as there's less heat to extract. But this is inapplicable to RV water heaters which don't operate with a heat pump.