That's all based on Shannon entropy (probabilistic), not Kolmogorov complexity (algorithmic), but there are a lot of connections between them. This paper is a pretty thorough summary: https://homepages.cwi.nl/~paulv/papers/info.pdf
"We define the cross-complexity of an object x with respect to another object y as the amount of computational resources needed to specify x in terms of y, and the complexity of x related to y as the compression power which is lost when adopting such a description for x, compared to the shortest representation of x"
Might not be exactly what you're asking, but Landauer's principles that information is physical and necessarily requires some energy to compute is relevant. The 'cost' Bennett is talking about here might be the energy required to change the entropy.
Also this treatment of K-L divergence as a measure of "Bayesian surprise": http://ilab.usc.edu/surprise/
That's all based on Shannon entropy (probabilistic), not Kolmogorov complexity (algorithmic), but there are a lot of connections between them. This paper is a pretty thorough summary: https://homepages.cwi.nl/~paulv/papers/info.pdf
And here's a paper defining algorithmic relative complexity by analogy to relative entropy: http://www.mdpi.com/1099-4300/13/4/902/pdf-vor
"We define the cross-complexity of an object x with respect to another object y as the amount of computational resources needed to specify x in terms of y, and the complexity of x related to y as the compression power which is lost when adopting such a description for x, compared to the shortest representation of x"