You can stipulate conditions to make the solution work out in either direction.

Suppose the bucket is the size of lake, and the leak is so miniscule that it takes many centuries to detect any loss. And also I need to keep the bucket full for a microsecond. In this case it is better to keep the bucket full, than to let it drain.

Now suppose the bucket is made out of chain-link and any water you put into it immediately falls out. The level is simply the amount of water that happens to be passing through at that moment. And also the next time I need the bucket full is after one century. Well in that case, it would be wasteful to be dumping water through this bucket for a century.

All heat that is lost must be replaced (we must input enough heat that the device returns to T_initial)

Hotter objects lose heat faster, so the longer we delay restoring temperature (for a fixed resume time) the less heat is lost that will need replacement.

Hotter objects require more energy to add another unit of heat, so the cooler we allow the device to get before re-heating (again, resume time is fixed) the more efficient our heating can be.

There is no countervailing effect to balance, preemptive heating of a device before the last possible moment is pure waste no matter the conditions (although the amount of waste will vary a lot, it will always be a positive number)

Even turning the heater off for a millisecond is a net gain.

Does it depend on whether you know in advance _when_ you need it back at the hot temperature?

If you don’t think ahead and simply switch the heater back on when you need it, then you need the heater on for_longer_.

That means you have to pay back the energy you lost, but also the energy you lose during the reheating process. Maybe that’s the countervailing effect?

> Hotter objects require more energy to add another unit of heat

Not sure about this. A unit of heat is a unit energy, right? Maybe you were thinking of entropy?