"It was the potentially dangerous subject of theology that concerned the Church far more than natural philosophy and most examples of discipline relate to the former. These systems were essentially the internal disciplinary procedures of the universities and, as we have seen, the usual sanction was little more than having to recant the error and amend ones work to correct it. Matters would only usually leave the auspices of the university if there was an appeal or if the matter became notorious and widely known as, for example, in the case of the Amalricians of Paris where the teaching of a university theologian threatened to produce a heretical sec"
"As many academics were also members of the mendicant orders, even in institution not nominally run by the Church so they were also under the governance of their order and could face disciplinary proceedings from this direction. The most famous case of this is Roger Bacon who appears to have been imprisoned by his superiors in the Franciscans for not having his work vetted by them before publication"
"The most infamous agents of medieval church discipline, the inquisitors, do not appear to have had a major role in dealing with academics but could become involved in certain cases. Word that someone had been teaching heretical opinions could reach the ear of the local inquisitor who would investigate and, on finding the allegations to be true, get the teacher to admit and recant his error before handing down his penance. As the inquisitor was not part of the university it is likely that the case would already have acquired a degree of notoriety, perhaps due to public disputations or lectures, before he heard of it and having done so he would be obliged to act."
"The facts are set out in the condemnation of Cecco who was burnt at the stake in Florence on 15th December 1327 [NOTE]. Three years previously, he had been found guilty of “utterances against the Catholic faith”, by the inquisitor Lambertus of Cingulo in Bologna where Cecco was a professor, with the result that he was fined, had his books confiscated and was banned from teaching or practising astrology. Unfortunately, the condemnation does not tell us what the offending utterances were, although later authorities, such as the fifteenth century inquisitor, Franciscus Florentinus, mention that he had taught and written that Jesus lived and suffered the way he did because he was born under a special star which had also led the magi from the East [NOTE]. Contrary to what Franciscus insists, Cecco does not mention any such thing in his extant books (even those that were burnt with him) so his utterances were in all likelihood verbal and made in lectures. As Cecco was not more severely punished we can also assume he confessed to and repented of his errors. However, his was clearly a serious heresy as he did not get away with a simple recantation such as required of Blasius of Parma in 1396 when he was also convicted of “utterances against the Catholic faith” [NOTE]. Cecco left Bologna and made his way to Florence where he promptly flouted the inquisitor’s strictures and became court astrologer to Jacob of Brescia. This wilful disobedience immediately marked him out as a recalcitrant heretic and when he found himself before the Florentine inquisitor, Accursius, it is no surprise that the he was handed over to the secular arm. As burning was the expected fate of a re-offender the judicial machinery seems to be working as expected."
What you need to remember is the Catholic church was in many ways central to life in Europe, and that many secular powers relied on the Church in some way to maintain their power. The power to declare someone a heretic was indeed powerful, and it was expected that ALL Catholic Rulers and Lords would enforce Church rulings within their holdings, both in regards to heretical rulings and other less serious offenses. Obviously they interfered most in Universities that either taught Theology, or in area's the Church had a vested interest in such as astronomy.
Another factor to remember is that the thing that made them Universities in the modern sense rather then the studium generale (university) and universitas (corporation of students or teachers) that preceded them was the issue of a Papal Bull which rendered them autonomous and provided the mechanism by which anyone admitted to be a teacher in the recognised university had the right to teach everywhere without further examinations.
Finally there was the concept of Studium generale or a registered institution of international excellence by the Holy Roman Empire, and these were considered the most prestigious places of learning in Europe. This may be part of receiving the Papal Bull, I'm not 100% sure if you could be issued a Papal Bull yet not be a Studium generale. Both theological and secular universities were registered. I believe the Vatican still does confer this title on Universities, but I'm not sure anyone really cares anymore.