Peter Turchin has tried to discover this kind of general rules (and actually to found a new field, “cliodynamics”) in his book Historical Dynamics: Why States Rise and Fall. I must admit that I am quite sceptical towards this kind of attempts, but this is probably the closest thing you will find to an actual statistic survey of an empire's life cycle. Like many similar new/fringe (I am using this word without any pejorative connotations) theories, cliodynamics rely a lot on the Internet for their diffusion, so you will probably be able to find pdf articles that summarize Turchin's main ideas on the topic.