He has NOT run before.
Well, I'm pretty sure he was seriously considering it - before he got sidelined. I'll try to source it.
I thought I saw it somewhere. So why did Arnold drop out?
http://www.enterstageright.com/archive/articles/0501tidbits1.htm Schwarzenegger drops bid to run for governor
Arnold Schwarzenegger is opting out of the 2002 race to become California's governor, saying his film career and family have taken precedence over politics.
"I have to be selfish at this point ... and take care of those things," the 53-year-old actor told the Los Angeles Times. "The movie projects came together. ... I have to keep up my end of the deal. It's not like it could have gone this way or that."
He said that he was not abandoning the idea of running for office as a Republican, but was only postponing his plunge into politics until his four children, ages 3 to 11, are older, the Times said April 25.
The Austrian-born former bodybuilder shares moderate views on gun control and abortion rights that match those of most California voters.
Schwarzenegger, whose film credits include "Pumping Iron," "Total Recall" and "End of Days," has two film projects in the works. He said he will focus on finishing a sequel to "True Lies" and production will begin later this year on "Terminator III."
Republican Secretary of State Bill Jones already has declared his candidacy and William E. Simon Jr., a Los Angeles investment banker, also is considering seeking the GOP nomination.
Incumbent Democrat Gray Davis, who likely will seek a second term, so far faces no serious threat from within his party.