  The extensive catalog of $\gamma$-ray selected flat-spectrum radio quasars
(FSRQs) produced by \emph{Fermi} during a four-year survey has generated
considerable interest in determining their $\gamma$-ray luminosity function
(GLF) and its evolution with cosmic time. In this paper, we introduce the novel
idea of using this extensive database to test the differential volume expansion
rate predicted by two specific models, the concordance $\Lambda$CDM and $R_{\rm
h}=ct$ cosmologies. For this purpose, we use two well-studied formulations of
the GLF, one based on pure luminosity evolution (PLE) and the other on a
luminosity-dependent density evolution (LDDE). Using a Kolmogorov-Smirnov test
on one-parameter cumulative distributions (in luminosity, redshift, photon
index and source count), we confirm the results of earlier works showing that
these data somewhat favour LDDE over PLE; we show that this is the case for
both $\Lambda$CDM and $R_{\rm h}=ct$. Regardless of which GLF one chooses,
however, we also show that model selection tools very strongly favour $R_{\rm
h}=ct$ over $\Lambda$CDM. We suggest that such population studies, though
featuring a strong evolution in redshift, may nonetheless be used as a valuable
independent check of other model comparisons based solely on geometric
considerations.
