The Kolmogorov complexity of a deterministic universe is constant. The Kolmogorov complexity of a nondeterministic universe grows over time. It grows whenever something happens that is not predetermined by its laws of nature. E.g. randomness or free will.
Would it be possible to measure a difference? If the universe's information content grows, does that also increase its energy content?
Edit: By "complexity of a universe" I mean the amount of information required to simulate it up to some point in time.