  We study continuity, and lack thereof, of thermodynamical properties for
one-dimensional dynamical systems. Under quite general hypotheses, the free
energy is shown to be almost upper-semicontinuous: some normalised component of
a limit measure will have free energy at least that of the limit of the free
energies. From this, we deduce results concerning existence and continuity of
equilibrium states (statistical stability). Counterexamples to statistical
stability in the absence of strong hypotheses are provided.
