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We present a microscopic approach based on the many-body density matrix formalism for the study of exciton-phononinduced features in absorption spectra of carbon nanotubes with an arbitrary chiral angle and diameter. The introduction of excitonic wave functions already containing the influence of the Coulomb interaction allows us to focus on the contribution of phonons. We observe the formation of pronounced sidebands on both sides of the zero-phonon line. We study the behaviour of exemplary nanotubes belonging to the two semiconducting and one metallic family with respect to the spectral weight transfer from the zero-phonon line to the exciton-phonon sidebands at different temperatures.