Cooling it will help the yeast floculate out of solution and settle to the bottom.
A fresh sugar solution is going to be aerobic, yeast multiply by cell division in the presence of oxygen and food (sugar). The small amount lost from pouring off the liquid (after cooling) will quickly be replaced.
When the solution goes anerobic (no oxygen) they shift their mode of respiration and stop putting energy into reproduction. This is the state we are ultimately looking for because this is when they begin producing large amounts of co2.
By reusing yeast, you have a large population where the majority of the yeast has not yet gone into what is basically a hibernation phase, or similar to what brine shrimp eggs do (waiting for ideal conditions to reanimate). They yeast will start chowing down immediately as long as you don't shock them with solution that is too warm.
If you are saving yeast, you can get two containers going so you can truley wash the yeast keeping the other in the fridge until you need to get it ready right before you take the other one off line.
I've found bread yeast simple enough to use. I should break out the hydrometer and see how much alcohol it is actually withstanding and how much of the sugar it is actually consuming.
I pour out the liquid and leave some of the settled yeast behind and pour some new yeast on top. It starts up quicker for me. I get CO2 within a couple of hours. I'm not concerned about the flavors in the final product, so a bacterial infection (wild yeast or bacteria that is not of the desired strain) is of no concern. A hot water rinse of the bottle is of little use to prevent that anyway. If it smells off when I dump it out or there is a drop in production, I'll start fresh and sanitize the bottle.
I usually feel kind of guilty using the quick reply... my replies are rarely quick.