What do you mean by "energy metric"?

Something that works as a measure over the state variables for the purposes of Lyapunov's stability theorems. That is, take a set of state variables that completely define the system, and take some measure that is always non-negative and is an increasing function of every variable. (Lyapunov's theorem -- one version -- says that if you can find such a measure, and if it's strictly non-increasing with time, the system is stable -- but this isn't guaranteed from the definition.)

Maybe you can find one that increases with out bound, but I don't know what energy metric you have in mind.

What I mean by having no 2nd law of thermodynamics is that it's possible to construct a Universal Turing machine that can operate indefinitely without using up any irreplaceable resources.

What does that have to do with the 2nd law? There are (physically possible) reversible computers that use no irreplaceable resources, so the fact that something is a Turing machine operating indefinitely does not mean that the 2nd law is being violated.

(I should also point out that Life defines time as an additional property of the universe, rather than a measure on the other properties, which is how time works in this universe. If you carry our universe's manifestation of time, and check whether that phenomenon exists in Life, it's not obvious that it does.)