Entropy of canonical ensemble - invariant wrt energy shift?

Main Question or Discussion Point

I have a short question which I have been discussing with a fellow student and a professor. The question (which is not a homework question!), is as follows:

If you shift all the energies $E_i \to E_i + E_0$ (thus also shifting the mean energy $U \to U + E_0$), does the entropy of the system remain the same?

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