||We use the newly developed finite temperature Lanczos algorithm to calculate the finite temperature properties of quantum spin models. To test the validity of this algorithm, we calculate the dynamical structure factor of the one-hole t-J model on a 16-site square lattice exactly, and compare it to the approximate result obtained by the finite temperature Lanczos method. We find that the Lanczos result reproduces the qualitative features of the exact dynamical structure factor satisfactorily. We then turn to the kagome antiferromagnet where we are interested in the low temperature behaviour of the specific heat capacity. Before we apply it to larger system, we test its validity on a 12-site system where the specific heat capacity can be calculated exactly. We find that it reproduces the exact result well. We then apply this method to a 21-site system, where exact evaluation is not feasible. We find that the high temperature peak is almost the same as in the 12-site system while the low temperature feature is different and becomes a shoulder. This is consistent with the previously found even-odd asymmetry in the system size. But our results show that this shoulder becomes more prominent as the system size increases, and may even have a tendency to develop into a peak.