I'm going through Elias Bickerman's The God of the Maccabees (Leiden: E.J. Brill, 1979) right now. He states the following:
"[W]hereas athletic contests were merely strange to Cappadocian custom, they were objectionable according to Jewish law...These high priests, then, one minute watched the exercises of the naked ephebes in the palaestra 'directly below the citadel,' and the next, climbed the steps of the altar to offer the sacrifice, adorned with the princely golden crown over the tiara, while the bells attacked to the gowns of the sons of Aaron rang, 'to make music as they walked' (Ecclus. 45:9). We must try to realize for ourselves the juxtaposition of these scenes, in order to make clear that, what was natural to the Greeks and an abomination to the Hebrews, could be seen in Jerusalem year after year. No schism occurred because of this, either in Jerusalem or in the diaspora. No new Pinchas (I Macc. 2:26) raised his hand against the transgressors of the law, and no indignant revolt broke out among the people" (41-42).
I'd like to know more. What's Bickerman source for his claim that athletic contests were objectionable according to Jewish law? Why did Jews deem it abominable for priests to be in the gymnasium once minute, then at the temple the next minute? Unfortunately, he doesn't offer documentation here. Maybe he will later in the book.