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We consider the modified Ising model introduced by de Oliveira, Mendes, and Santos [J. Phys. A 26, 2317 (1993)], where the temperature depends locally on the spin configuration and detailed balance and local equilibrium are not obeyed. We derive a relation between the linear response function and correlation functions that generalizes the fluctuation-dissipation theorem. In the stationary states of the model, which are the counterparts of the Ising equilibrium states, the fluctuation-dissipation theorem breaks down due to the lack of time reversal invariance. In the nonstationary phase-ordering kinetics, the parametric plot of the integrated response function vs the autocorrelation function is different from that of the kinetic Ising model. However, splitting into a stationary and an aging term , we find , and a numerical value of consistent with aχ= 14 , as in …