  Improving matrix-product state techniques based on the purification of the
density matrix, we are able to accurately calculate the finite-temperature
dynamic response of the infinite spin-1 XXZ chain with single-ion anisotropy in
the Haldane, large-$D$ and antiferromagnetic phases. Distinct thermally
activated scattering processes make a significant contribution to the spectral
weight in all cases. In the Haldane phase intraband magnon scattering is
prominent, and the onsite anisotropy causes the magnon to split into singlet
and doublet branches. In the large-$D$ phase response, the intraband signal is
separated from an exciton-antiexciton continuum. In the antiferromagnetic
phase, holons are the lowest-lying excitations, with a gap that closes at the
transition to the Haldane state. At finite temperatures, scattering between
domain-wall excitations becomes especially important and strongly enhances the
spectral weight for momentum transfer $\pi$.
