Monte Carlo simulation of model proteins on a cubic lattice are used to study the thermodynamics and kinetics of protein folding over a wide range of temperatures. Both random sequences and sequences designed to have a pronounced minimum of energy are examined. There is no indication in the kinetics of a "glass" transition at low temperature, i.e., below the temperature of the equilibrium folding transition, the kinetics of folding is described by the Arrhenius law at all temperatures that were examined. The folding kinetics is single-exponential in the whole range of studied temperatures for random sequences. The general implications of the temperature dependence of the folding rate are discussed and related to certain properties of the energy spectrum. The results obtained in the simulations are in qualitative disagreement with the conclusions of a theoretical analysis of protein folding kinetics based on certain kinetics assumptions introduced in the Random Energy Model. The origins of the discrepancies are analyzed and a simple phenomenological theory is presented to describe the temperature dependence of the folding time for random sequences.