I actually let my lye come down to between 100F and ambient and it seems fine. My experience with stearic acid says do keep the oils warm. I try to keep them around 170F.
Also, here are the molds I mentioned (forgot to take pics of them when I was doing shaving soap, but they work for any pucks of course)
Used for CP, green clay swirl
View attachment 38505
Used for an HP bath and body soap
View attachment 38506
These are technically ABS pipe, but PVC works just as well. If you own, or have access to, a miter saw you can easily cut any lengths you like. A light sanding easily removed any burrs. Currently I am just using uncoated parchment paper as a liner, which works well when double layered. I may try a sheet of acetate instead at some point in order to avoid the wrinkly edges on the HP pucks. I did pad the caps with cardboard as it makes them much easier to remove when they are not pressed on as far.
Of course you can also just put your shaving soap directly into a jar or tin or whatnot