Plumbing – Should I turn off the cold water while replacing a hot water faucet stem

Chances are either the valve seal has disintegrated and clogged the line, or other sediment has broken free and done the same.

Disassembling a faucet is usually fairly simple. Remove the set screw retaining the handle, then start removing the exposed nut, washers, and valve components. Lay them out in order for correct reassembly. Check for damage and debris.

If your valves are not soldered in (if they're installed using compression fittings), replacement is fairly simple. It's a good opportunity for even an occasional DIYer to get his hands dirty. Post a photo or accurate description for more assistance.

There could be multiple reasons. In order of likeliness.
As a safety feature there are anti scald features of those faucets. Roman style tubs move a high amount of water and very possible there is a mixing valve upstream that is opening when the water is hot. Very quickly the hot water is evacuated and the mixing valve closes. Slowly heats up and the cycle repeats Or Incorrect cartridge in the tub faucet handles. Perhaps they installed the hot cartridge on the cold set and vice verse.

Or

Poor insulation on the piping but this would give te symptom of a long amount of cold then warm with eventual hot water.

I'd say rhe first solution is most likely. You can try and fill the tub at a slow rate with both handles open. The hot more than the cold.

Either way if you can find the mixing valve. Perhaps near the jacuzzi access panel or a nearby access box in the wall or under a sink you can possibly adjust the controls to allow higher temperatures. Keeping the mixing valve from opening as often. But this may not be so in an apartment building it could very well be in a mechanical room under he buildings maintenance control.

Also the other faucets don't exhibit this due to he flow control aerators keeping flow rates down in the 1gpm range depending on local code.