  Exploring the origin of the pseudogap is important for the understanding of
superconductivity in cuprates. Here we report a systematical experimental study
on the phonon vibrational properties of
Tl$_2$Ba$_2$Ca$_{n-1}$Cu$_n$O$_{2n+4+\delta}$ ($n$=1,2,3) single crystals based
on the Raman scattering measurements over the temperature range from 10 to 300
K. The temperature evolution of the frequency and linewidth of the observed
phonon modes in each member of this family does not follow the expected
self-energy effect when entering the superconducting state. Instead, these
phonon modes exhibit a universal softening behavior below the temperature
around 150 K, which is higher above the superconducting transition. From the
comparison with the existing experimental data for various orders, we find that
the observed starting temperature for the phonon softening corresponds to the
onset opening temperature of the pseudogap. This finding indicates a large
lattice effect in the pseudogap state and the non-negligible spin-phonon
coupling for such a phonon softening.
