  We study the Chern-Simons number diffusion rate in the (1+1)-dimensional
latticeAbelian Higgs model at temperatures much higher than, as well as
comparable to, the sphaleron energy. It is found that in the high-temperature
limit the rate is likely to grow as power 2/3 of the temperature. In the
intermediate-temperature regime, our numerical simulations show that very weak
temperature dependence of the rate, found in previous work, persists at smaller
lattice spacings. We discuss possibilities of relating the observed behavior of
the rate to static finite-temperature properties of the model.
