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I vaguely recall that Talmud Avodah Zarah mentions a prohibition of eating gevinat akum - cheese manufactured by an idol worshipper (as I understand this definition.)

If there already was a prohibition against chalav akum (milk handled by an idol worshipper), why was there an additional need for gevinat akum. Cheese is made from milk.

Rambam Hilchos Machalos Asuros 3:12-13

חלב בהמה טמאה, אינו נקפה ועומד כחלב הטהורה; ואם נתערב חלב טהורה בחלב טמאה--כשתעמיד אותו--יעמוד חלב הטהורה, וייצא חלב הטמאה עם הקוס של גבינה. ומפני זה ייתן הדין שכל חלב הנמצא ביד גוי אסור, שמא עירב בו חלב בהמה טמאה; וגבינת הגויים מותרת, שאין חלב בהמה טמאה מתגבן. אבל בימי חכמי משנה גזרו על גבינת הגויים, ואסרוה מפני שמעמידין אותה בעור הקיבה של שחיטתן שהיא נבילה

The milk of a non-kosher animal will not congeal and solidify as the milk of a kosher animal does. If the milk of a non-kosher animal is mixed together with the milk of a kosher animal, when the mixture is [set aside for cheese to be made], the kosher milk will solidify and the non-kosher milk will be expelled together with the whey of the cheese.

Accordingly, logic would dictate that any milk found in the possession of a gentile is forbidden, lest the gentile have mixed the milk of a non-kosher animal with it. And the cheese of the gentiles should be permitted, for the milk of a non-kosher animal will not form cheese. Nevertheless, during the age of the Sages of the Mishnah, they issued a decree against gentile cheese and forbade it, lest they use the skin of the stomach of an animal they slaughtered - which is forbidden as a nevelah - to cause it to solidify.

Translation from Chabad.org. See also footnote 29 there.

Tosafos Avodah Zarah 39b, translation by Daf Yomi Advancement Forum

ואע״ג דקא חשיב חלב הוצרך לחשוב גבינה

Despite the fact that milk was mentioned, cheese also had to be mentioned. (Aren't they forbidden for the same reason? Why would both have to be mentioned?)

לפי שאין טעם איסורן שוה שחלב אסור משום עירוב חלב טמא אבל טעם גבינה הוא משום גילוי כדפרי' לעיל (דף לה.)

This is because the reason for their prohibition is not the same. Milk is forbidden because we suspect that unkosher milk will be mixed in to the kosher milk, while cheese is only forbidden because it is left out uncovered, as we explained earlier (35a, Tosfos DH "Chada").

Shakh to Yoreh Deah 115:20 quotes a number of opinions (notably Mordechai and Maharam MiRottenburg) which forbid gevinat akum even where a Jew watched the whole process from the start of the milking.

So in such a case the milk would be considered chalav yisrael, but nevertheless the fact that a non-Jew was the one to make cheese from it means that the cheese is forbidden under the separate rubric of gevinat akum.

Milk and gevina are appearenlty two issues. (translations from Sefsria).

In Mishna AZ 2.6. there is a mention of Milk of Akum.

אלו דברים של גוים אסורין ואין איסורן איסור הנאה, חלב שחלבו גוי ואין ישראל רואהו, והפת, והשמן שלהן. רבי ובית דינו התירו בשמן. ‏

Milk not watched is prohibited only for consumption. The Chashash is perhaps milk of non kasher species is mixed.

But for Gvina, AZ 2.4-5

המורייס וגבינת בית אונייקי של גויים--אסורין, ואיסורן איסור הניה, דברי רבי מאיר; וחכמים אומרין, אין איסורן איסור הניה.

The fish-brine and Bithynian cheese of non-Jews are prohibited and it is prohibited to derive any benefit from them, according to Rabbi Meir. The Sages say: Their prohibition is not against deriving benefit from them.

ב,ה אמר רבי יהודה, שאל רבי ישמעאל את רבי יהושוע כשהיו מהלכין בדרך, מפני מה אסרו את גבינת הגויים. אמר לו, מפני שמעמידין אותה בקיבת נבילה. אמר לו, והלוא קיבת העולה חמורה מקיבת הנבילה, ואמרו כל כוהן שדעתו יפה שורפה חיה; ולא הודו לו, אלא אמרו לא נהנין ולא מועלין. אמר לו, מפני שמעמידין אותה בקיבת עבודה זרה. אמר לו, אם כן, למה לא אסרוה בהניה. והשיאו לדבר אחר; אמר לו, ישמעאל אחי, היאך אתה קורא "כי טובים דודיך מיין" (שיר השירים א,ב [the proof is from the next words על כן עלמות אהבוך]), או כי טובים דודייך מיין. אמר לו, כי טובים דודייך מיין. אמר לו, אין הדבר כן, שהרי חברו מלמד עליו, "לריח שמניך טובים" (שיר השירים א,ג).‏

Rabbi Yehuda says: Rabbi Ishmael asked Rabbi Yehoshua [a question] as they were walking along the road. He said to him, "What is the cause for the prohibition against the cheese of non-Jews?" He said to him, "Because they curdle it inside the stomach of carrion." He said to him, "But is not [the law regarding] the stomach of a burnt offering more stringent than the stomach of carrion?! And they [the Sages] said: [It was proposed that] a priest with a good disposition may burn [a burnt offering after sucking out the fat from its stomach] while it it still raw [and has not yet been burnt with the offering, which would forbid one form deriving any benefit from it]. And they [the sages] did not agree with him [who proposed this opinion], but they said: One may not derive benefit [from the fats of the stomach], nor is one [who does so] liable for meilah [deriving prohibited benefit from from a sanctified object]." [Implying that one should therefore not be liable for the less stringent case of a stomach of carrion.] He [Rabbi Yehoshua] retracted, and [instead] said, "Because they curdle it in the stomachs of calves that were used for idolatry." And he [Rabbi Ishmael] said, "If so, why is there no prohibition to benefit from it?" He [Rabbi Yehoshua] redirected him to another topic. He said to him, "Ishmael, my brother, how do you read (Shir HaShirim 1): 'For dodechah [Heb. masc: "your love"] is better than wine,' or 'For dodayich [Heb. fem: "your love") is better than wine'? He said to him:, "'For dodayich is better than wine.'" He said to him, "The matter is not so. For its fellow [the following verse] teaches about it: 'For the fragrance of shemaneikha [Heb. masc: "your oils"] is good.'"

Bartenura says

שם כפר, שרוב עגלים הנמצאים שם קרבים לעבודה זרה. ר' מאיר חייש למיעוטא וסבר דאע"ג דרוב עגלים הוו מיעוטא כנגד שאר בהמות, חיישינן למיעוטא, וכל הגבינות הנמצאות שם אמרינן דלמא בקיבת עגלי ע"ז העמידן. וחכמים לא חיישי למיעוטא. ואין הלכה כר' מאיר בכל הני תלתא בבי: ‏

In Mishna 4. There was a place in which a part of livestock, a minority were dedicated to Idolatry. There was a Chashash according to Rabbi Meir, who thinks that the din is safek rennet of idolatry despite the minority of idolatry. But there is an Isur to eat them.

What is the prohibition?

In Mishna 5, From the conclusion of the Gemara explained in Rambam and Bartenura comments, the problem is rennet of nevela, stomacal tissue from a non shchited animal. This content is really not a part of the body, it's as excrement, as it was proven by Rabbi Yehoshua. This is a Chashash not elucidated in the Mishna. But as mefarshim explains, it was an official law decree. And for this Rabbi Yishmael keep the secret about the true reason of the secret until its broadcast. After the broadcast, the secret was revealed. The Chashash is that the rennet is gastric tissue, that is nevela mamash.

We see from the sources that that was a separate gzera for cheese.
For Halacha I don't know precisely what is the din nowadays that there isn't objectively doubt about non kasher animals milk. Some poskim allowed milk. If a Jew make cheese with this milk they certainly allow it. But to buy from goym cheese seems to be prohibited.