If the week old pumpkin pie has been refrigerated and has no mold or bad smell to it then yes you can eat week old pumpkin pie.
I have kept pumpkin pies in the refrigerator for a week and it's been okay.
You can also freeze pumpkin pie to make it last longer.
You can put a hot pumpkin pie in the fridge if you want too and it will be just fine.
Although it will make your refrigerator come on and stay on longer so that the refrigerator can cool down the heat from the pumpkin pie.
It's recommended that you allow the pumpkin pie to cool down for at least 30 minutes before putting the pumpkin pie into the refrigerator.
I've also left pumpkin pies out overnight and they've been fine even though it's best to refrigerate the pumpkin pie after cooking it if you don't plan to eat it or serve it within the same day or within a few hours.
Leaving a pumpkin pie out overnight is fine and the pumpkin pie will still remain safe to eat the next day without refrigeration.
I've heard before that you should refrigerate the pumpkin pie when not eating or serving it to prevent bacteria from growing on the pumpkin pie.
Pumpkin pies are usually made with eggs, milk and cream which can go bad when left out longer than a few hours.
However while it's recommended to refrigerate pumpkin pie when not eating it within a few hours the pumpkin pie will remain safe to eat when left out overnight.