  Thermodynamic properties of the SU($n$) Heisenberg model in one dimension is
studied by means of high-temperature expansion for arbitrary $n$. The specific
heat up to $O[(\beta J)^{23}]$ and the correlation function up to $O[(\beta
J)^{18}]$ are derived with $\beta J$ being the antiferromagnetic exchange in
units of temperature. It is found for $n>2$ that the specific heat shows a
shoulder in the high-temperature side of a peak. The origin of this structure
is clarified by deriving the temperature dependence of the correlation
function. With decreasing temperature, the short-range correlation with
two-site periodicity develops first, and then another correlation with $n$-site
periodicity at lower temperature. This behavior is in contrast to that of the
inverse square interaction model, where the specific heat shows a single peak
according to the exact solution. Our algorithm has an advantage that neither
computational time nor memory depends on the multiplicity $n$ per site; the
series coefficients are obtained as explicit functions of $n$.
