Heat capacity data near the smectic-A-hexatic-B transition in thin freestanding liquid-crystal films has been obtained. Films greater than two molecular layers in thickness display multiple anomalies associated with the smectic-A-hexatic-B transition. The anomalies are sharply peaked and are therefore inconsistent with recent defect mediated melting theory. Moreover, the separate surface and interior transition data can be successfully fit to a simple power law. The fitting results indicate that these films exhibit similar, continuous transitions at significantly different temperatures localized to regions only 30 Angstrom apart.