The temperature dependence of third sound is determined from a variati
onal-density-matrix approach. At zero temperature a liquid He-4 monola
yer, adsorbed to a weakly attractive substrate, supports third sound o
nly within a restricted range of coverages. Elevating the temperature
appreciably decreases the value of the minimum stable coverage, increa
ses the maximum sound velocity, and slightly decreases the maximum sta
ble coverage.