Colder is better from a sanitization perspective.
I would probably crash cool as best you can, decant some of the start wort, and pitch the rest. A one gallon starter needs time to properly crash cool and flocculate. Visually if the starter wort is cloudy, there's still yeast in suspension. You'll have to make a judgement call.
Alternatively, you can pitch the entire starter volume. As I stated earlier, one gallon of spent starter wort is a little much for me, so I would avoid adding that to my beer. If I make my starter far enough in advance, I tend to decant the majority of the starter wort. Starter wort isn't supposed to taste good, so I try to add as little as possible to my beer. For me, it's just less junk, less variables, in my finished beer. Sometimes you do whatcha gotta do.