  We consider the possibility that dark matter halos are described by the
Fermi-Dirac distribution at finite temperature. This is the case if dark matter
is a self-gravitating quantum gas made of massive neutrinos at statistical
equilibrium. This is also the case if dark matter can be treated as a
self-gravitating collisionless gas experiencing Lynden-Bell's type of violent
relaxation. In order to avoid the infinite mass problem and carry out a
rigorous stability analysis, we consider the fermionic King model. In this
paper, we study the non-degenerate limit leading to the classical King model.
This model was initially introduced to describe globular clusters. We propose
to apply it also to large dark matter halos where quantum effects are
negligible. We determine the caloric curve and study the thermodynamical
stability of the different configurations. Equilibrium states exist only above
a critical energy $E_c$ in the microcanonical ensemble and only above a
critical temperature $T_c$ in the canonical ensemble. For $E<E_c$, the system
undergoes a gravothermal catastrophe and, for $T<T_c$, it undergoes an
isothermal collapse. We compute the profiles of density, circular velocity, and
velocity dispersion. We compare the prediction of the classical King model to
the observations of large dark matter halos. Because of collisions and
evaporation, the central density increases while the slope of the halo density
profile decreases until an instability takes place. We show that large dark
matter halos are relatively well-described by the King model at, or close to,
the point of marginal microcanonical stability. At that point, the King model
generates a density profile that can be approximated by the modified Hubble
profile. This profile has a flat core and decreases as $r^{-3}$ at large
distances, like the observational Burkert profile. Less steep halos are
unstable.
