At least 40 times. There are deep scouring marks on the inside bottom at Gibralter.
I suspect that the Mediterranean was somewhat lower (and stable) at a point prior to the end of the Ice Age.
When the 'break-through' ocurred at Gibralter and the Med began filling swamping the many islands...there would have been refugees everywhere...coastal towns would have been flooded also. It would be enough to make a person believe that the whole world was flooding and rush to build a big boat way up on one of the Ararat mountains.
When the break-though ocurred at the Bosphorus, it just confirmed that the end was near.
BTW, I also suspect that the mummies from the Tarim Basin may have been refugees from all this flooding...they were pushed way out to the margins of liveable areas.
These refugees probably spread farming and the Indo-European languages across greater Europe too.