  The ionisation cones of NGC5728 have a deficit of molecular gas based on
millimetre observations of CO(2-1) emission. Although photoionisation from the
active nucleus may lead to suppression of this transition, warm molecular gas
can still be present. We report the detection of eight mid-infrared rotational
H$_2$ lines throughout the central kiloparsec, including the ionisation cones,
using integral field spectroscopic observations with JWST/MIRI MRS. The H$_2$
line ratios, characteristic of a power-law temperature distribution, indicate
that the gas is warmest where it enters the ionisation cone through disk
rotation, suggestive of shock excitation. In the nucleus, where the data can be
combined with an additional seven ro-vibrational H$_2$ transitions, we find
that moderate velocity (30 km s$^{-1}$) shocks in dense ($10^5$ cm$^{-3}$) gas,
irradiated by an external UV field ($G_0 = 10^3$), do provide a good match to
the full set. The warm molecular gas in the ionisation cone that is traced by
the H$_2$ rotational lines has been heated to temperatures $>200$ K. Outside of
the ionisation cone the molecular gas kinematics are undisturbed. However,
within the ionisation cone, the kinematics are substantially perturbed,
indicative of a radial flow, but one that is quantitatively different from the
ionised lines. We argue that this outflow is in the plane of the disk, implying
a short 50 pc acceleration zone up to speeds of about 400 km s$^{-1}$ followed
by an extended deceleration over $\sim$700 pc where it terminates. The
deceleration is due to both the radially increasing galaxy mass, and
mass-loading as ambient gas in the disk is swept up.
