  We show that for stochastic dynamical systems out of equilibrium the
violation of the fluctuation-dissipation equality is bounded by a function of
the entropy production. The result applies to a much wider situation than `near
equilibrium', comprising diffusion as well as glasses and other macroscopic
systems far from equilibrium. For aging systems this bounds the age-frequency
regimes in which the susceptibilities satisfy FDT in terms of the rate of decay
of the H-function, a question intimately related to the reading of a
thermometer placed in contact with the system.
