We study the thermodynamics and critical behavior of su (m ) spin chains of Haldane-Shastry type at zero chemical potential, both in the AN −1 and BCN 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 ≲10 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 at 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.