We report on a study of the heat capacity of thin films of 4He adsorbed in porous Vycor glass for temperatures between 20 and 700 mK. As in previous work, we find a critical coverage, nc∼26.6μmoles/m2, above which we find superfluidity. For coverages greater than nc, the heat capacity varies linearly with temperature above the superfluid transition temperature Tc. We also observe a linear dependence at high temperatures for coverages below nc. At lower temperatures, however, the heat capacity for these coverages decreases with a much stronger temperature dependence. We discuss these data in the context of recent theoretical work on the onset of superfluidity in disordered systems.