A microscopic approach is proposed to the damping of the double giant dipole resonance ͑DGDR͒. The double-time Green's function method is used to derive a closed set of coupled equations for the propagation of two-phonon excitation through the field of incoherent nucleon pairs. The analytical expressions for the width and energy shift of the DGDR are obtained. The numerical calculations are performed for 90 Zr, 90 Sn, and 208 Pb for several characteristics of the DGDR at zero as well as nonzero temperatures T. The results are found in reasonable agreement with existing experimental systematics for the width and energy of the DGDR. As compared to the estimation within the harmonic picture, the anharmonicity leads to a noticeable enhancement of the integrated photoabsorption cross section ͑IPACS͒ over the DGDR region. The DGDR width is found to increase sharply with increasing T at Tр3 MeV, but goes to a saturation at T Ͼ3 MeV. The harmonic limit for the DGDR width is restored already at Tу1.5 MeV. It is shown that the IPACS of the DGDR can also be enhanced compared to its harmonic value if it is built on a hot GDR
