  We discuss the low-temperature specific heat of the integrable SU(N)-
invariant Heisenberg model in one dimension with degrees of freedom in the
symmetric rank-$m$ tensor representation, especially for the antiferromagnetic
coupling. It is known that the linear specific heat coefficient $\gam$ is a
function of a field which breaks the SU(N) invariance of the internal degrees
of freedom. We calculate the $\gam$ in a zero field and in a small field. The
in-field $\gam$ is less than the zero-field $\gam$ as expected since the
entropy is reduced in the ordered system. The zero-field $\gam$ is the same as
the one obtained by the prediction of the critical behavior of 1D quantum spin
systems via conformal field theory. This extends the previous results for N=2
to an arbitrary N.
