  Microcanonical thermodynamics studies the operations that can be performed on
systems with well-defined energy. So far, this approach has been applied to
classical and quantum systems. Here we extend it to arbitrary physical
theories, proposing two requirements for the development of a general
microcanonical framework. We then formulate three resource theories,
corresponding to three different sets of basic operations: i) random reversible
operations, resulting from reversible dynamics with fluctuating parameters, ii)
noisy operations, generated by the interaction with ancillas in the
microcanonical state, and iii) unital operations, defined as the operations
that preserve the microcanonical state. We focus our attention on a class of
physical theories, called sharp theories with purification, where these three
sets of operations exhibit remarkable properties. Firstly, each set is
contained into the next. Secondly, the convertibility of states by unital
operations is completely characterised by a majorisation criterion. Thirdly,
the three sets are equivalent in terms of state convertibility if and only if
the dynamics allowed by theory satisfy a suitable condition, which we call
unrestricted reversibility. Under this condition, we derive a duality between
the resource theory of microcanonical thermodynamics and the resource theory of
pure bipartite entanglement.
