  We study the form factors of local operators of integrable QFT's between
states with finite energy density. These states arise, for example, at finite
temperature, or from a generalized Gibbs ensemble. We generalize Smirnov's form
factor axioms, formulating them for a set of particle/hole excitations on top
of the thermodynamic background, instead of the vacuum. We show that exact form
factors can be found as minimal solutions of these new axioms. The
thermodynamic form factors can be used to construct correlation functions on
thermodynamic states. The expression found for the two-point function is
similar to the conjectured LeClair-Mussardo formula, but using the new form
factors dressed by the thermodynamic background, and with all singularities
properly regularized. We study the different infrared asymptotics of the
thermal two-point function, and show there generally exist two different
regimes, manifesting massive exponential decay, or effectively gapless behavior
at long distances, respectively. As an example, we compute the few-excitations
form factors of vertex operators for the sinh-Gordon model.
