I'm struggling to understand what you're asking here.
As I understand it, the 'arrow of time' is simply a philosophical explanation for why processes happen one way but don't immediately undo themselves. We postulate that entropy can never decrease; so that's a constraint on the passage of time - that processes can't happen in 'reverse' as seen from our perspective, because you'd have to get rid of some entropy, which can't be done.
However, there's no such constraint on entropy. You can - in theory - put energy back if you take it away, and vice versa - there's no explicit law that restricts the 'direction' of energy flow. Energy can be constrained by means of entropic variation - and almost always is in practice, but that's a property of entropy, not energy.
In other words, because energy has a conservation law, things can 'run in either direction' because you end up with the same amount of stuff as you started with. Entropy isn't conserved and has a law that defines how it changes, so that produces the 'arrow of time' effect.