Symptoms sound very similar to my problem, which turned out to be an AC relay pack (apparently it's 4 relays all together.) Sorry the receipt from my mechanic doesn't list a part number but he said this is something he was able to purchase from a local Toyota dealer, so it's a Toyota part.
2012 Porsche 991.1S white
2007 Exige S arctic silver (project
2005 Elise magnetic blue/biscuit
2014 Audi Q5 3.0T scuba blue