sort of a grinding noise when pushing start...

Yes! I had exactly this noise in our 2012 Prius v the day before yesterday on cold start. It was an alarmingly loud sound of metal grinding on metal. I turned the power off because it sounded like the engine was being damaged. It did it again a few minutes later so I took it to the dealership (noise stopped after 10-15 seconds). The only thing they could find was a loose heat shield contacting the exhaust manifold. We just picked it up tonight, so we'll see if it makes the noise again tomorrow morning. I'm really skeptical about that being the problem, however, as this was not a rattle -- it was an extremely loud clunking, clanging, grinding noise. The only time I've heard anything like it was when our old Audi engine timing chain broke, ending its short (but expensive) life. I'm extremely worried about the Prius v because it has 33,000 miles on it. We bought a Toyota because we didn't want to deal with the VW/Audi car repairs anymore. Not happy!

 

I think Diana is talking about a different noise to yours. Yours is probably the electric brake pump building up hydraulic pressure ready for action, can't say I've ever heard it on mine after switching off though.
Does it do it when you get in the car before you press the start button ?

 

The vacuum pump can run before, during or after the car is started. Opening the door will set it off too.

 

I had exact same problem. It was a cold day, about 24 F and I heard noise somewhere from the engine. I ended up turned it off and didn't drive that day. Next day, it happened again, but noise was gone after a minute later. I took it to dealership and spent its night over there. Pretty weird, they didn't find any problem or any noise (as I don't expect anything happen to my car, but there was). Technician said he called Toyota (!?) and they acknowledge the problem but currently don't have a fix yet!?
Didn't have good experience with my dealership at all. They held my card without providing me any transportation. I had to transport myself back and forward to work and to the dealership. Unlike Acura dealership, they let me used their vehicle in the meantime, or at least took me home and pick up later when it done.

 

Same problem here. (Prius II) first time at 9 degrees F, then this morning at 49 degrees F within a week. Car starts fine with no noise and after about 5 seconds of running there is a loud grinding sound from under the hood for about 3 seconds, the master warning light comes on and stays on for about 5 minutes. It's happened twice so tomorrow it goes in to the Stealership.