Starting from a generalized nonequilibrium entropy, and from an extension of the Einstein formula for the probability of the fluctuations, we obtain the second moments of nonequilibrium fluctuations of the specific energy and the heat flux in rigid heat conductors. This approach leads to nonequilibrium corrections to the usual fluctuation-dissipation expressions for the heat fluctuations. The corresponding numerical corrections are found for phonon heat transport in dielectric solids and for electronic heat transport in metals. The macroscopic results share common features with those obtained for heat transport in harmonic chains by an information-theoretical approach.