We calculate the thermal partition functions of open strings on the S-brane
backgrounds (the bouncing or rolling tachyon backgrounds) both in the bosonic
and superstring cases. According to hep-th/0302146, we consider the discretized
temperatures compatible with the pure imaginary periodicity of tachyon
profiles. The ``effective Hagedorn divergence'' is shown to appear no matter
how low temperature is chosen (including zero-temperature). This feature is
likely to be consistent with the large rate of open string pair production
discussed in hep-th/0209090 and also emission of closed string massive modes
hep-th/0303139. We also discuss the possibility to remove the divergence by
considering the space-like linear dilaton backgrounds as in hep-th/0306132.Comment: 33 pages, no figure; v2 typos corrected, a reference adde
