We calculate the temperature dependence of NMR relaxation rate and electrical resistivity for coupling to a local, strongly anharmonic phonon mode. We argue that the two-phonon Raman process is dominating NMR relaxation. Due to the strong anharmonicity of the phonon an unusual temperature dependence is found having a low temperature peak and becoming constant towards higher temperatures. The electrical resistivity is found to vary like T 2 at low temperatures and following a √ T behavior at high temperatures. Both results are in qualitative agreement with recent observations on β-pyrochlore oxide superconductors.PACS numbers: 74.70. Dd, 74.25.Kc The recent discovery of superconductivity in the family of pyrochlore oxides KOs 2 O 6 , RbOs 2 O 6 , and CsOs 2 O 6 has attracted great interest because of their unusual properties. Among these KOs 2 O 6 with the highest T c of 9.6 K appears to be the most unusual. The temperature dependence of its electrical resistivity shows a strong concave-downward temperature dependence [1,2], in contrast to the other two compounds, where a T 2 temperature dependence at low temperatures has been observed [3,4]. Specific heat measurements have shown a large mass enhancement, large specific heat jump at T c , and existence of low frequency Einstein modes [5,6]. Bandstructure calculations have indicated that the anomalies, in particular in KOs 2 O 6 , might be due to a highly anharmonic low frequency rattling motion of the alkali-ion inside an oversized cage formed by the Os and O ions [7,8]. This is consistent with X-ray observations of anomalously large atomic displacements for the K ions [9] and low frequency phonon structures seen in photoemission spectroscopy [10].Recent observations of NMR relaxation rates 1/T 1 T at the K site have been demonstrated to be entirely dominated by the vibrations of the K ion via coupling of the electric field gradient to the nuclear quadrupole moment [11]. Such a domination of phonons for nuclear relaxation usually occurs in diamagnetic insulators, but is extremely rare in metals and was attributed to the rattling motion of the K ions. The temperature dependence of 1/T 1 T was found to be anomalous as well, showing a peak around 12-14 K and decreasing at higher temperatures. It has been argued that such a behavior is inconsistent with the two-phonon Raman process, which usually dominates quadrupolar relaxation, and it has been interpreted in terms of the direct phonon process with a strongly temperature dependent phonon damping rate [11]. In the superconducting state 1/T 1 T exhibits a sudden decrease, suggesting a strong coupling of the phonon mode to the conduction electrons and an associated increase of the phonon lifetime in the superconducting state.Motivated by these experimental findings, in the present work we study the influence of a local, strongly anharmonic, and damped phonon mode on the NMR relaxation rate and the electrical resistivity. We find that within this model the two-phonon Raman process is expected to dominate NMR relaxati...