Summary: Let \(H(q,p)=\frac{1}{2}{{p}^{2}}+V(q)\) be a 1-degree of freedom mechanical Hamiltonian with a \(C^r\) periodic potential \(V\) where \(r>4\). The Nosé-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 Nosé’s. These results complement the result of F. Legoll et al. who proved the existence of such tori near the decoupling limit [Arch. Ration. Mech. Anal. 184, No. 3, 449–463 (2007; Zbl 1122.82002); Nonlinearity 22, No. 7, 1673–1694 (2009; Zbl 1173.37066)].