There is a way (somewhat handwaved) in which someone can travel through time. This is not merely the standard one dimension of time, for there are branching timelines. Whenever you travel into the past, you being in the past causes the timeline to diverge into separate timelines. They do not, however, necessarily grow further and further apart. Distinctions between temporal dimensions and dimensions of space are rather arbitrary from a non-human perspective, and gravity does not only work across three dimensions, a force that pulls mass together gradually over a period of time, but also pulls universes of separate timelines together.
The effect that this has is that a temporal force is exerted, causing properties of adjacent universes to trend towards each other. The closer the universes are to each other, physically, i.e. the less extreme the differences are, the greater the 'temporal force' that causes the timelines to converge.
Convergence does neither add nor remove energy from the universe; a force is merely something that changes one form of energy to another.
There are several theories in-universe on what will eventually happen to timelines as a whole. These are similar to those of the fate of the universe we have now. Theories include: All timelines will eventually converge, similar to 'the big crunch' theory; some timelines will diverge indefinitely, getting further and further away from each other as time goes on; eventually, all timelines will remain equidistance away from each other, and be parallel.
Any energy transferred across timelines will be gradually replaced in the form of entropy being transferred as a result of what are basically homeostasis reactions.
Are there any current unexplained mysteries of physics which would be explained by the discovery of this kind of temporal gravity?