  We consider the Fermi-Pasta-Ulam-Tsingou (FPUT) chain composed by $N \gg 1$
particles and periodic boundary conditions, and endow the phase space with the
Gibbs measure at small temperature $\beta^{-1}$. Given a fixed ${1\leq m \ll
N}$, we prove that the first $m$ integrals of motion of the periodic Toda chain
are adiabatic invariants of FPUT (namely they are approximately constant along
the Hamiltonian flow of the FPUT) for times of order $\beta$, for initial data
in a set of large measure. We also prove that special linear combinations of
the harmonic energies are adiabatic invariants of the FPUT on the same time
scale, whereas they become adiabatic invariants for all times for the Toda
dynamics.
