  The classical definition of the virial temperature of a galaxy halo excludes
a fundamental contribution to the energy partition of the halo: the kinetic
energy of non-thermal gas motions. Using simulations from the FOGGIE project
(Figuring Out Gas & Galaxies In Enzo) that are optimized to resolve low-density
gas, we show that the kinetic energy of non-thermal motions is roughly equal to
the energy of thermal motions. The simulated FOGGIE halos have $\sim 2\times$
lower bulk temperatures than expected from a classical virial equilibrium,
owing to significant non-thermal kinetic energy that is formally excluded from
the definition of $T_\mathrm{vir}$. We derive a modified virial temperature
explicitly including non-thermal gas motions that provides a more accurate
description of gas temperatures for simulated halos in virial equilibrium.
Strong bursts of stellar feedback drive the simulated FOGGIE halos out of
virial equilibrium, but the halo gas cannot be accurately described by the
standard virial temperature even when in virial equilibrium. Compared to the
standard virial temperature, the cooler modified virial temperature implies
other effects on halo gas: (i) the thermal gas pressure is lower, (ii)
radiative cooling is more efficient, (iii) O VI absorbing gas that traces the
virial temperature may be prevalent in halos of a higher mass than expected,
(iv) gas mass estimates from X-ray surface brightness profiles may be
incorrect, and (v) turbulent motions make an important contribution to the
energy balance of a galaxy halo.
