It was shown by Tim Austin that if an orbit equivalence between probability-measure-preserving actions of finitely generated amenable groups is integrable then it preserves entropy. I will discuss some joint work with Hanfeng Li in which we show that the same conclusion holds for the maximal sofic entropy when the acting groups are countable and sofic and contain an amenable w-normal subgroup which is not locally virtually cyclic, and that it is moreover enough to assume that the Shannon entropy of the cocycle partitions is finite (what we call Shannon orbit equivalence). It follows that two Bernoulli actions of a group in the above class are Shannon orbit equivalent if and only if they are conjugate. I will also describe a topological version of our measure entropy invariance result, along with an application to the construction of simple C*-simple groups whose von Neumann algebras have property Gamma.