  We theoretically investigate the thermodynamic properties of a strongly
correlated two-dimensional Fermi gas with a confinement-induced negative
effective range of interactions, which is described by a two-channel model
Hamiltonian. By extending the many-body T-matrix approach by Nozi\`eres and
Schmitt-Rink to the two-channel model, we calculate the equation of state in
the normal phase and present several thermodynamic quantities as functions of
temperature, interaction strength, and effective range. We find that there is a
non-trivial dependence of thermodynamics on the effective range. In experiment,
where the effective range is set by the tight axial confinement, the
contribution of the effective range becomes non-negligible as the temperature
decreases down to the degenerate temperature. We compare our finite-range
results with recent measurements on the density equation of state, and show
that the effective range has to be taken into account for the purpose of a
quantitative understanding of the experimental data.
