Indeed, though the Elvis tune isn't on the soundtrack album. It's also worth pointing out that if one wants to get really technical, there are several seconds of original Williams scoring (for the Paramount logo / "mountain" transition) prior to the appearance of Elvis.
But then again, the specific version of "Anything Goes" used in Temple of Doom was done for that movie - not just the performance / recording, but the arrangement (and even a lyrics translation), whereas "Hound Dog" is just the classic vintage track plunked right into place; "Anything Goes" does even incorporate some fleeting moments of original scoring in its own beginning (and blurring the diegetic and non-diegetic elements, for a fusion of score / source).
But still again, there's also plenty of original Williams music in both scores, both on the albums and as-yet-unreleased. Both are highly thematic and clearly "Indy" scores, though both are also quite characteristic of Williams' scores in general at those two different points in his career. They can certainly be compared and contrasted any number of ways, but in the end both have their own distinctive strengths, as well as a few they share. There's no reason one can't enjoy one, the other, both (or neither!), and one's personal preferences really don't require justification - if one likes one score more than the other, both equally, or whatever, then so be it.