  We investigate the relic density n_\chi of non-relativistic long-lived or
stable particles \chi in non-standard cosmological scenarios. We calculate the
relic abundance starting from arbitrary initial temperatures of the
radiation-dominated epoch, and derive the lower bound on the initial
temperature T_0 \geq m_\chi/23, assuming that thermally produced \chi particles
account for the dark matter energy density in the universe; this bound holds
for all \chi annihilation cross sections. We also investigate cosmological
scenarios with modified expansion rate. Even in this case an approximate
formula similar to the standard one is capable of predicting the final relic
abundance correctly. Choosing the \chi annihilation cross section such that the
observed cold dark matter abundance is reproduced in standard cosmology, we
constrain possible modifications of the expansion rate at T \sim m_\chi/20,
well before Big Bang nucleosynthesis.
