  We use a high-statistic quantum Monte Carlo and Maximum Entropy
regularization method to compute the dynamical energy correlation function
(DECF) of the one-dimensional (1D) $S=1/2$ antiferromagnetic Heisenberg model
at finite temperatures. We also present a finite temperature analytical ansatz
for the DECF which is in very good agreement with the numerical data in all the
considered temperature range. From these results, and from a finite temperature
generalisation of the mechanism proposed by Lorenzana and Sawatsky [Phys. Rev.
Lett. {\bf 74}, 1867 (1995)], we compute the line shape for the optical
absorption spectra of multimagnon excitations assisted by phonons for quasi 1D
compounds. The line shape has two contributions analogous to the Stokes and
anti-Stokes process of Raman scattering. Our low temperature data is in good
agreement with optical absorption experiments of CuO$_3$ chains in
Sr$_2$CuO$_3$. Our finite temperature results provide a non trivial prediction
on the dynamics of the Heisenberg model at finite temperatures that is easy to
verify experimentally.
