score:6
Found information and more or less contemporary sources on Google Books (here, here and here, for example) and in gallica.fr (here), but no source cares to say how he died, only hinting that it was sudden and unexpected; the only person I found who said something about how he died was Voltaire, who, in his letters, didn't hide his rejoicing at Royer's death (they had a quarrel before):
Then the Grand Turk, our ambassador to the Ottoman Porte, and Royer, have all died of an indigestion? I'm very angry about M. des Alleurs, whom I loved, but I find solace in the loss of Royer and the Grand Turk.
and later on:
A stroke has punished Royer for disfiguring my verses; it's up to me now to take care of my prose.
EDIT- I'm adding extra information:
The "Grand Turk" was the Ottoman Sultan Mahmud I, who died in December 1754 – his death was sudden, too, and the 1911 Encyclopaedia Britannica says he died from heart disease, so it's easy to infer he had a stroke; the ambassador was Roland Puchot, the mentioned (Count) des Alleurs,not Ailleurs as I wrote before (corrected that), who died in Constantinople in late 1754 or early 1755, of a stroke ("un attaque d'apoplexie" according to this source).
So, if the sources and Voltaire are to be believed, Pancrace Royer has most probably died of a stroke.