  We find the minimum and the maximum value for the local energy of an
arbitrary finite bipartite system for any given amount of entanglement, also
identifying families of states reaching these bounds and sharing formal
analogies with thermal states. Then, we numerically study the probability of
randomly generating pure states close to these energy bounds finding, in all
the considered configurations, that it is extremely low except for the
two-qubit and highly degenerate cases. These results can be important in
quantum technologies to design energetically more efficient protocols.
