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When I compare the usages of "above the threshold" to those of "over the threshold" in Google Scholar, in both cases I find a multitude of highly-cited prestigious papers which used either of them.
So, are they both correct, and do they interchangeably mean the same thing?
In many cases, these phrases have the same meaning and can safely be used interchangeably. But that is not quite always the case. First, there is a specific meaning, for example: "In many cultures it is traditional for a newly-wed husband to carry his bride over the threshold of their new home." It would be quite strange in that case to use "above", and most native speakers would think it wrong - but this is a usage which is quite specific. In this case, "across the threshold" would also be normal.
More usually, you'll be using the phrase to say that something has exceeded some level which causes or allows some effect. In this case, either phrase can often be used. However, in technical language "above" is more usual and sometimes seems to be essential. As an example, "The brightness of the new comet is expected to rise over the threshold for naked-eye visibility by the weekend." If "above" were used, the meaning is the same, is equally understandable, and seems a little more natural. In a formal or official letter or paper, "above" would almost always be used: "The noise level in the factory is above the threshold where it can cause hearing damage and must be reduced immediately."
