Etymologists/Linguists posit that the prefix for gold in Proto-Indo-European was:
*aus-
Which gave way to *auzom in Proto-Italic languages and ausum in Classical Latin.
At some point, ausum morphed into aurum.
What would be the cause of the change from /s/ to /r/, considering the phonemes are very different?
This is a sound change called rhotacism, one of the better-attested Latin sound changes.
Back in Old Latin times, we see a phoneme /s/ showing up all over the place. For example, the stem for "flower" is flos-, which means its nominative is flos-s (which becomes flos), its genitive is flos-is, and its accusative is flos-em. And the infinitive is regularly formed with a suffix -se (extrapolated from an old S-stem noun), as in es-se, amā-se, amāvis-se.
Somewhere in there, this phoneme got voiced to [z] when it appears between two vowels and isn't geminated. This was still written with the letter S.
But then, this [z] got lenited to [ɹ], an approximant like the R-sound in Midwestern American English. This was the same as the letter the Romans wrote R at the time, so the two merged together. Later, this merged sound turned into a trill [r]. (Or maybe R was already a trill, and [z] merged into it later. It's hard to tell.)
So now what happened to those words? Suddenly the paradigms became less regular: flos, but floris, florem; esse, amāvisse, but amāre. This is presumably also what happened to ausum: it became aurum.
Then, to fill in the gap left by the vanishing S, a double /ss/ got shortened to /s/ after long vowels. So now S can appear between vowels again, and you get the state of affairs seen in Classical times.
(Note that all phonetics given here are speculative, since we have no recordings of native speakers of Old Latin to compare against. But they're relatively confident speculations.)
