  Integrable quantum spin chains display distinctive physical properties making
them a laboratory to test and assess different states of matter. The study of
the finite temperature properties is possible by use of the thermodynamic Bethe
ansatz, however at the expense of dealing with non-linear integral equations
for, in general, infinitely many auxiliary functions. The definition of an
alternative finite set of auxiliary functions allowing for the complete
description of their thermodynamic properties at finite temperature and fields
has been elusive. Indeed, in the context of $sl_n$-invariant models
satisfactory auxiliary functions have been established only for $n \leq 4$. In
this paper we take a step further by proposing a systematic approach to
generate finite sets of auxiliary functions for $sl_n$-invariant models. We
refer to this construction as the canonical formulation. The numerical
efficiency is illustrated for $n=5$, for which we present some of the
thermodynamic properties of the corresponding spin chain.
