  We study the thermodynamics and critical behavior of su($m$) spin chains of
Haldane-Shastry type at zero chemical potential, both in the $A_{N-1}$ and
$BC_N$ cases. We evaluate in closed form the free energy per spin for arbitrary
values of $m$, from which we derive explicit formulas for the energy, entropy
and specific heat per spin. In particular, we find that the specific heat
features a single Schottky peak, whose temperature is well approximated for
$m\lesssim10$ by the corresponding temperature for an $m$-level system with
uniformly spaced levels. We show that at low temperatures the free energy per
spin of the models under study behaves as that of a one-dimensional conformal
field theory with central charge $c=m-1$ (with the only exception of the
Frahm-Inozemtsev chain with zero value of its parameter). However, from a
detailed study of the ground state degeneracy and the low-energy excitations,
we conclude that these models are only critical in the antiferromagnetic case,
with a few exceptions that we fully specify.
