We present measurements of the superfluid density and heat capacity of thin films of He4 adsorbed in aerogel glass. We find that the critical behavior of the superfluid density for films with transition temperatures between 50 mK and 1 K is similar to that seen for films adsorbed on Vycor. Power-law behavior is observed over at least one order of magnitude in reduced temperature with an exponent ζ=0.63±0.03. This exponent is much smaller than the value 0.811±0.004 found when the pores were filled with helium. We also find that the exponent is unchanged by the addition of small amounts of He3. These results are consistent with the existence of a correlation length that diverges at the superfluid transition. Unlike for full-pore He4 aerogel, the correlation length is long compared with all structural length scales throughout the superfluid density power-law regime, thus leading to the more bulklike critical behavior. The heat capacity of He4 films in aerogel shows no singularity like that seen for full-pore He4 aerogel or for thin films adsorbed in Vycor. We show that this result is consistent with the predictions of hyperuniversality given the long correlation lengths implied by the superfluid density measurements.