  We investigate the effect of thermal fluctuations on the two-particle
spectral function for a disordered $s$-wave superconductor in two dimensions,
focusing on the evolution of the collective amplitude and phase modes. We find
three main effects of thermal fluctuations: (a) the phase mode is softened with
increasing temperature reflecting the decrease of superfluid stiffness; (b)
remarkably, the non-dispersive collective amplitude modes at finite energy near
${\bf q}=[0,0]$ and ${\bf q}=[\pi,\pi]$ survive even in presence of thermal
fluctuations in the disordered superconductor; and (c) the scattering of the
thermally excited fermionic quasiparticles leads to low energy incoherent
spectral weight that forms a strongly momentum-dependent background halo around
the phase and amplitude collective modes and broadens them. Due to momentum and
energy conservation constraints, this halo has a boundary which disperses
linearly at low momenta and shows a strong dip near the $[\pi,\pi]$ point in
the Brillouin zone.
