The implications of thermodynamics of heteropolymers for their folding kinetics are formulated and discussed. The predictions are tested by Monte Carlo simulation of folding of a lattice model 36-monomer proteins at different temperatures. Using a simulated annealing procedure in sequence space, a number of sequences are designed which have sufficiently low energy in a given target conformation. This conformation plays a role of the native structure for the model proteins. The folding transition is found to be cooperative, and the nature of the free energy barrier is studied. At high temperature the barrier is entropic, and at low temperature it is mainly energetic. This can be explained by transformation of the early partly folded intermediate from the disordered state (belonging to the quasicontinuous part of the energy spectrum) at high temperature to the non-native low-energy frozen conformation at low temperature. The latter plays a role of an off-pathway trap. A parallel folding process has been detected at low temperature where direct folding competes with relaxation of the intermediate. The frozen intermediate must unfold to make it possible to form a folding nucleus, which is the prerequisite of subsequent fast descent to the native state.