  A temperature behavior of the frustrated translational mode (T-mode) of a
light particle, coupled by different regimes of ohmicity to the surface, is
studied within a formalism of the generalized diffusion coefficients. The
memory effects of the adsorbate motion are considered to be the main reason of
the T-mode origin. Numerical calculations yield a thermally induced shift and
broadening of the T-mode, which is found to be linear in temperature for Ohmic
and super-Ohmic systems and nonlinear for strongly sub-Ohmic ones. We obtain
analytical expressions for the T-mode shift and width at weak coupling for the
systems with integer "ohmicity" indexes n=0-2 in zero temperature and high
temperature limits. We provide an explanation of the experimentally observed
blue- or red-shifts of the T-mode on the basis of a comparative analysis of two
typical times of the system evolution: a time of decay of the
"velocity-velocity" autocorrelation function, and a correlation time of the
thermal bath random forces. A relation of the T-mode to the multiple jumps of
the adsorbate is discussed, and generalization of conditions of the multiple
hopping to the case of quantum surface diffusion is performed.
