  We theoretically investigate the high-temperature thermodynamics of a
harmonically trapped Fermi gas across a narrow Feshbach resonance, by using the
second-order quantum virial expansion, and point out some new features compared
to the broad resonance. The interatomic interaction is modeled by the
pseudopotential with an additional parameter, i.e., the effective range, to
characterize the narrow resonance width. Deeply inside the width of a narrow
Feshbach resonance, we find the second virial coefficient evolves with the
effective range from the well-known universal value 1/4 in the broad-resonance
limit to one another value 1/2 in the narrow-resonance limit. This means the
Fermi gas is more strongly interacted at the narrow resonance. In addition, far
beyond the resonance width, we find the harmonically trapped Fermi gas still
manifests appreciable interaction effect across a narrow Feshbach resonance,
which is contrary to our knowledge of the broad Feshbach resonance. All our
results can be directly tested in current narrow Feshbach resonance
experiments, which are generally carried out in a harmonic trap.
