In this paper, the jump transition characteristic of sunspot cycle length(henceforth SCL) from 522BC upward were analyzed. Moving t-test method based results indicate that there were no appreciable change of SCL in recent 2500 years. However, the variances of SCL series(henceforth Sv) show apparent jump transitions in 260BC, 31AD, 271AD, 1172AD and 1844AD, respectively. When Sv jump from low to high level, the "cold" climatic jumps were occurred, and vice versa. Moreover, the highest level of Sv coincide with the frequently occurrence of natural disaster in the Little Ice Age, which indicates that the instability of climate was caused by the unstable SCL(solar activity).