  Let H(q,p) = p^2/2 + V(q) be a 1-degree of freedom mechanical Hamiltonian
with a C^n periodic potential V where n>4. The Nos\'e-thermostated system
associated to H is shown to have invariant tori near the infinite temperature
limit. This is shown to be true for all thermostats similar to Nos\'e's. These
results complement the result of Legoll, Luskin and Moeckel who proved the
existence of such tori near the decoupling limit.
