I'm in the "face lather only, mugs are for porridge blah blah" camp.
And I'm also in the cool/cold/whatever comes out of the tap camp, it just lowered the irritation level and improved the shave results.

But before I went cold...I found that the brushes that take up a lot of water in soaking (here Piggy-Piggy) did a good job of staying "warmish" if soaked in hot water.

Leatherstockings has a good summary of what they would call in consultant speak "best practices for ____ "

And give the cold/cool water a try...you shave with a Straight Razor so it's not like they are going to come for your Man Card any time...ever...