Not necessarily. Consider the time-turner in HPMOR. You could have physics which allow such stable time loop, with no determinism on which loop among the possible ones will actually occur, and yet have conservation of energy.
As I mentioned a few times, HPMoR time turners violate general relativity, as they result in objects appearing and disappearing without any energy being extracted from or dissipated into the environment. E.g. before the loop: 1 Harry, during the loop: 2 Harries, after the loop: 1 Harry.
This is a thread where people can ask questions that they would ordinarily feel embarrassed for not knowing the answer to. The previous thread is at close to 500 comments.