I have a copy of The History of Love that someone sent to me on PBS. I was going to list it here, but all of a sudden I noticed there was no price anywhere on the book and at the top of the back cover it says "Advance Praise for The History of Love". Would you assume this is an advanced reader copy? I dont see anything else indicating it is an advanced copy and the ISBN matches that on PBS. Advice?

If it doesn't indicate advanced reader copy, it's probably not.  It'd have words like "not for resale", "reviewers please check text against final edition", "book release date x/xx/xx", etc.  This sounds like it's not an ARC.  The advance praise is most likely from people that were given advanced readers copies and their blurbs were then printed on the back cover to entice you to buy the book.

If it's a hardback, not having a price could mean it's a book club edition. I've never seen an ARC come in hardback.

Paperbacks that are ARCs will say so - and also will not have a bar code on it!