Consider an FPU chain composed of particles, and endow the phase space with the Gibbs measure corresponding to a small temperature . Given a fixed , we construct packets of normal modes whose energies are adiabatic invariants (i.e., are approximately constant for times of order , ) for initial data in a set of large measure. Furthermore, the time autocorrelation function of the energy of each packet does not decay significantly for times of order . The restrictions on the shape of the packets are very mild. All estimates are uniform in the number of particles and thus hold in the thermodynamic limit , .