  We study thermal broadening of the hole spectral function of the
two-dimensional t-J model (and its extensions) within the non-crossing
approximation with and without the contribution of optical phonons. We find
that phonons at finite temperature broaden the lowest energy quasiparticle
peak, however, the string excitations survive even for relatively strong
electron-phonon coupling. Experimental angle resolved photo-emission
spectroscopy(ARPES) results compare well with our calculations at finite
temperature when we use strong electron-phonon coupling without any adhoc
broadening. In addition, we have studied the role of vertex corrections and we
find that their contribution allows us achieve the same overall agreement with
the ARPES experimental results but using smaller values for the electron-phonon
coupling.
