  We demonstrate that astrophysical constraints on the dense-matter equation of
state place an upper bound on the color-superconducting gap in dense matter
above the transition from nuclear matter to quark matter. Pairing effects in
the color-flavor locked (CFL) quark matter phase increase the pressure at high
density, and if this effect is sufficiently large then the requirements of
causality and mechanical stability make it impossible to reach such a pressure
in a way that is consistent with what is known at lower densities. The
intermediate-density equation of state is inferred by considering extensions of
chiral effective field theory (CEFT) to neutron star densities, and
conditioning these using current astrophysical observations of neutron star
radius, maximum mass, and tidal deformability (PSR J0348+0432, PSR J1624-2230,
PSR J0740+6620, GW170817). At baryon number chemical potential $\mu =
2.6~\text{GeV}$ we find a 95% upper limit on the CFL pairing gap $\Delta$ of
$457~\text{MeV}$ using overly conservative assumptions and $216~\text{MeV}$
with more reasonable assumptions. This constraint may be strengthened by future
astrophysical measurements as well as by future advances in high density QCD
calculations.
