  We construct an extensive adiabatic invariant for a Klein-Gordon chain in the
thermodynamic limit. In particular, given a fixed and sufficiently small value
of the coupling constant $a$, the evolution of the adiabatic invariant is
controlled up to times scaling as $\beta^{1/\sqrt{a}}$ for any large enough
value of the inverse temperature $\beta$. The time scale becomes a stretched
exponential if the coupling constant is allowed to vanish jointly with the
specific energy. The adiabatic invariance is exhibited by showing that the
variance along the dynamics, i.e. calculated with respect to time averages, is
much smaller than the corresponding variance over the whole phase space, i.e.
calculated with the Gibbs measure, for a set of initial data of large measure.
All the perturbative constructions and the subsequent estimates are consistent
with the extensive nature of the system.
