I remember hearing that a cohen must write his get numerous times: since today the get is being written by a sofer, I presume this custom is not in use anymore. But was there such a rule in the past? If yes, what is the source?
Not quite. They would only write one get but the writing and signing was made a complicated affair. Since a Kohen is not allowed to marry a divorcee, he couldn't remarry his former wife once the get was given. As Kohanim were known to be "hotheads", the danger was that they would give their wife a get on impulse, and later regret it. [Non Kohanim could also be hot-heads :-), but they could remarry their ex-wives afterwards]. In order to enable the Kohen to cool down, the Sages instituted a very much drawn-out procedure of the writing of the get. This involved writing one line, folding over the paper and stitching it down, with witnesses signing on the fold, then writing the second line, fold it over etc etc. So, the only thing that was done many times, was the signing of the witnesses. Today this isn't done anymore, although the reason for that isn't exactly clear. It has been suggested that the Sages only instituted it for their days. See Tosafot Bave Batra 160b "Tiknu" https://www.sefaria.org.il/Bava_Batra.160b.10?lang=he&p2=Tosafot_on_Bava_Batra.160b.10.1&lang2=he&w2=About&lang3=he Neither the Rambam, nor the Shulchan Aruch bring the laws of the so-called Get Mekushar. However see Rema Choshen Mishpat 42-1 & Shach ibid 3 [https://www.sefaria.org.il/Shulchan_Arukh%2C_Choshen_Mishpat.42.1?lang=he&with=all&lang2=he and https://www.sefaria.org.il/Shulchan_Arukh%2C_Choshen_Mishpat.42.1?lang=he&p2=Siftei_Kohen_on_Shulchan_Arukh%2C_Choshen_Mishpat.42.3.1&lang2=he