Phat is a slang spelling of fat, originating and common among African-Americans. It has come to the attention of standard English speakers relatively recently, but like many such slang terms of a subpopulation it is much older than those outside of the subgroup. Time magazine records it as early as 2 August 1963:

Negro argot...Mellow, phat, stone, boss. General adjectives of approval.

Given the source, we can safely assume that phat was in slang use at least as far back as the 1950s. Its recent appearance in mainstream publications and writing is largely due to crossover from Hip-Hip into mainstream American culture.

Phat is also a word that frequently is given a false acronymic origin. The exact acronym varies with the telling, Pretty Hot And Tempting, Pretty Hips And Thighs, and Pussy Hips Ass Tits have all been suggested. There is no evidence supporting any acronymic origin.

Rather, phat is most likely simply a slang respelling of fat. Such respellings are common in slang. And fat has a long history of meaning rich, abundant, or desirable. Fat has been used this way in English since the early 17th century, and in other languages for far longer. The specific sexual connotation of phat is likely just a specialization of the general meaning. Some suggest it may be a clipping of emphatic. Again, there is no evidence for this last, but at least it’s more plausible than any of the acronymic origins.

(Source: Oxford English Dictionary, 3rd Edition)

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