[Torkel]

Does it depend though?

The water (heat) leaking out is what you need to add back. As water level drops (hotend cools) the leaking will slow. So any replenishing means more leakage then you are eventually paying for by adding more water (heat) in.

[markisus]

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.

[bcoates]

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.