  The nonlinear response to an oscillating field is calculated for a kinetic
trap model with an exponential density of states and the results are compared
to those for the model with a Gaussian density of states. The calculations are
limited to the high temperature phase of the model. It is found that the
results are qualitatively different only in a temperature range near the glass
transition temperature $T_0$ of the exponential model. While for the Gaussian
model the choice of the dynamical variable that couples to the field has no
impact on the shape of the linear response, this is different for the
exponential model. Here, it is found that also the relaxation time strongly
depends on the variable chosen. Furthermore, the modulus of the frequency
dependent third-order response shows either a peak or exhibits a monotonuous
decay from a finite low-frequency limit to a vanishing response at high
frequencies depending on the dynamical variable. For variables that give rise
to a peak in the modulus it is found that its height either increases or
decreases as a function of temperature, again depending on the details of the
choice of the variable. The peak value of the modulus shows a scaling behavior
near $T_0$. It is found that for some variables the low-frequency limit of the
cubic response diverges at the glass transition temperature and also at a
further temperature determined by the particular variable. A recently proposed
approximation that relates the cubic response to a four-time correlation
function does not give reliable results due to a wrong estimate of the
low-frequency limit of the response.
