  Time-delay cosmography is a powerful technique to constrain cosmological
parameters, particularly the Hubble constant ($H_{0}$). The TDCOSMO
collaboration is performing an ongoing analysis of lensed quasars to constrain
cosmology using this method. In this work, we obtain constraints from the
lensed quasar WGD 2038-4008 using new time-delay measurements and previous mass
models by TDCOSMO. This is the first TDCOSMO lens to incorporate multiple lens
modeling codes and the full time-delay covariance matrix into the cosmological
inference. The models are fixed before the time delay is measured, and the
analysis is performed blinded with respect to the cosmological parameters to
prevent unconscious experimenter bias. We obtain $D_{\Delta t} =
1.68^{+0.40}_{-0.38}$ Gpc using two families of mass models, a power-law
describing the total mass distribution, and a composite model of baryons and
dark matter, although the composite model is disfavored due to kinematics
constraints. In a flat $\Lambda$CDM cosmology, we constrain the Hubble constant
to be $H_{0} = 65^{+23}_{-14}\, \rm km\ s^{-1}\,Mpc^{-1}$. The dominant source
of uncertainty comes from the time delays, due to the low variability of the
quasar. Future long-term monitoring, especially in the era of the Vera C. Rubin
Observatory's Legacy Survey of Space and Time, could catch stronger quasar
variability and further reduce the uncertainties. This system will be
incorporated into an upcoming hierarchical analysis of the entire TDCOSMO
sample, and improved time delays and spatially-resolved stellar kinematics
could strengthen the constraints from this system in the future.
