Anomalous room-temperature specific heats have been seen experimentally by Knapp and coworkers in highT c superconductors. The measurements depart from 3N k B +γ0T by an amount of orderγ0T. We present a theory for this, based on the fact that electronphonon interactions cause both electron band energiese k n and phonon frequenciesω Q µ to be weakly temperature-dependent. First an observation is made which we call Brook's Theorem: the shiftE k n (Qµ) of the electron energye k n due to adding one phonon in modeQµ, is identical to the shiftΔΩ Q µ (kn) of the phonon energyωQµ due to adding one electron in the statekn. Second, we assert that to sufficient accuracy whenT>Θ D , entropy of the system of interacting electrons and phonons equals the entropy which would be calculated for non-interacting particles, except that theT-dependent energiesEkn(T) andω Q µ (T) should be used. A proof to orderu2 (whereu is phonon displacement amplitude) was given in the thesis of one of us. Third, we use Brook's Theorem to show that the correctionsΔ Cel andΔ Cph to the specific heat due toT-dependence of electron and phonon energies respectively, are identically equal and have order of magnitudeγ0T. All these results are valid in adiabatic approximation to orderu2, and fail at low temperatures where non-adiabatic corrections are large.