  We combine the results of perturbative continuous unitary transformations
with a mean-field calculation to determine the evolution of the single-mode,
i.e., one-triplon, contribution to the dynamic structure factor of a two-leg
$S=1/2$ ladder on increasing temperature from zero to a finite value. The
temperature dependence is induced by two effects: (i) no triplon can be excited
on a rung where a thermally activated triplon is present; (ii) conditional
excitation processes take place if a thermally activated triplon is present.
Both effects diminish the one-triplon spectral weight upon heating. It is shown
that the second effect is the dominant vertex correction in the calculation of
the dynamic structure factor. The matrix elements describing the conditional
triplon excitation in the two-leg Heisenberg ladder with additional four-spin
ring exchange are calculated perturbatively up to order 9. The calculated
results are compared to those of an inelastic neutron scattering experiment on
the cuprate-ladder compound La$_{4}$Sr$_{10}$Cu$_{24}$O$_{41}$ showing
convincing agreement for established values of the exchange constants.
