  Context. The birth environments of planetary systems are thought to influence
planet formation and orbital evolution, through external photoevaporation and
stellar flybys. Recent work has claimed observational support for this, in the
form of a correlation between the properties of planetary systems and the local
Galactic phase space density of the host star. In particular, Hot Jupiters are
found overwhelmingly around stars in regions of high phase space density, which
may reflect a formation environment with high stellar density. Aims. We instead
investigate whether the high phase density may have a galactic kinematic
origin: Hot Jupiter hosts may be biased towards being young and therefore
kinematically cold, because tidal inspiral leads to the destruction of the
planets on Gyr timescales, and the velocity dispersion of stars in the Galaxy
increases on similar timescales. Methods. We use 6D positions and kinematics
from Gaia for the Hot Jupiter hosts and their neighbours, and construct
distributions of the phase space density. We investigate correlations between
the stars' local phase space density and peculiar velocity. Results. We find a
strong anticorrelation between the phase space density and the host star's
peculiar velocity with respect to the Local Standard of Rest. Therefore, most
stars in "high-density" regions are kinematically cold, which may be caused by
the aforementioned bias towards detecting Hot Jupiters around young stars
before the planets' tidal destruction. Conclusions. We do not find evidence in
the data for Hot Jupiter hosts preferentially being in phase space
overdensities compared to other stars, nor therefore for their originating in
birth environments of high stellar density.
