We report theoretical results for heat transport by quasiparticle excitations in superfluid He infused into silica aerogel engineered with uniaxial anisotropy. For this system, two distinct equal spin pairing (ESP) superfluid phases have been reported based on NMR spectroscopy. Theoretical analysis predicts the first ESP state to be the chiral A phase with chiral axis aligned along the strain axis, and the lower temperature phase to be a polar-distorted chiral phase with random transverse fluctuations in the orientation of the chiral axis. We report calculations of heat transport for the high temperature chiral phase, including an anomalous (zero field) thermal Hall current originating from branch conversion scattering of Bogoliubov quasiparticles by the chiral order parameter induced by potential scattering by the silica aerogel. Observation of an anomalous thermal Hall current would provide a direct signature of the underlying chirality and topology of the superfluid phase of He in “stretched” silica aerogels.