Can I cook with cosmetic coconut oil?
Yes, I think you can.  The page you link does list the oil as "food grade", and specifies no added solvents... so there's probably nothing dangerous in the jar and it can be safely used for cooking.

In general, though, you usually shouldn't equate cosmetic treatments and edible substances, even if some of the ingredients are the same (soaps often do this, list all food sounding ingredients until a second glance reveals it, well, soap), or even if the base substance is the same but may have undergone different processes (like coconut oil or cocoa butter).  If the coconut oil had been processed with some non-food grade solvents or processes, it might be perfectly safe for topical application, and still be quite dangerous to consume.

If you are questioning whether a substance might be an ingredient as well as a cosmetic treatment, look up the information, and I mean brand-specific product page, and see if it mentions "food grade" - that means exactly the bit of information you're looking for, that it is of a high enough quality that it passes the tests they use for product meant to be eaten.