I have seen multiple posts on this page that explained the statistical definition of Temperature as the derivative of the Entropy to the energy:

\begin{equation} \frac{1}{T}\equiv \left(\frac{\partial S}{\partial U}\right)_{V, N} \end{equation} Where $S$ has always been the Boltzmann Entropy,

$$ S = - k_B \ln \Omega(U, V, N) $$

My Question is: Does this Definition also apply to other Ensembles than the micro-canonical Ensemble? For any other Ensemble, can I also calculate $ \dfrac{\partial S}{\partial \langle U\rangle }$ where $\langle U\rangle $ is supposed to be the averaged Energy and $S = -k_B \langle \ln(\rho)\rangle$ is not any-more the Boltzmann entropy, but instead the Gibbs - Entropy, with $\rho$ being the probability distribution of the system?

I know that there are ensembles in which temperature is given externally and then defines the state of the system, like the canonical ensembles. But you can still assume for a given set of equilibrium states, that the temperature "changes" with the change of average Energy $\langle U\rangle$, and thus my question is up also for those cases.


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