Everything is in the title:
How often do Math.SE questions appear on the 'Hot questions' SE section?
This depends on exactly what you mean. I'll try to give a broad overview. This is a subject that I've been doing an extensive amount of data mining on for fun, so I happen to have a great deal of info on it, but for now I'm just stating the basics.
In my answer here, I gave a list of 53 questions which topped the network hot questions list over the past 146 days. That's an average of one such question every ~2.75 days. These are only the ones that reach the top of the list when I captured data on this. Anything that was only on top of the list for a few minutes, or which was on top of the list only for a period of time which I was offline and not collecting data, is excluded. Hence, the real number is probably slightly more frequent.
Those questions which did make the top of the list typically stay there for a significant period of time. In my data, MSE questions were on top of the list 32% of all the times I took data. This isn't a number I'm terribly satisfied with, since my data was not collected at regular intervals (I'm not online 24 hours a day). Hence, there could be significant biases just from time-of-day effects. There's also a great deal of variation on how long individual questions stay on top, ranging from a few minutes to over a week, so one shouldn't take it as any more than a reasonable estimate. If we go by this estimate, the average time one of these 53 questions stayed on top of the list is about 21 hours. I think the median is probably only a few hours, but there is a significant long tail of questions staying hot for over a week.
If you mean the full list of questions, then things get a bit harder. The full network hot questions list has 100 questions. I've only ever took data on the top 50 because the top 100 are quite varied and many only stay there for a few minutes. It's not impossible for a question with only one answer and one or two upvotes on the question and answer to make it into the top 50, or even top 30. This is especially true during off-hours when most American and European visitors are sleeping. During these hours new questions are pretty rare network-wide, and the ones that do show up have a good shot at making it on the list.
The median number of questions from Math SE in the top 50 network-wide in my data was 4. The average was ~4.7. The number ranged from 0 (quite rare; accounted for ~1% of the time) to 11 at the highest. Unfortunately I don't think I can come up with a solid estimate for how long such questions last, but it's typically on the order of a few hours for most of the ones that don't continue to rise to the top of the list.
All of these numbers are quite high compared to other network sites. Math is by far the most common site on the hot questions list. Part of that is the sheer number of questions which come in on this site, over 500 per day (#3 is only 200 per day). That's 2nd only to SO, but SO seems to have much more conservative voting tendencies and also doesn't typically allow very basic questions. In practice SO only rarely makes it on the list, though the ones that do usually make it to the top and stay there for quite a while and accumulate a huge number of votes. I think SO questions may also have some penalty in the formula. Another part is the answering and voting tendencies of the community here, which reward even basic questions with more answers and upvotes. That's exactly what the algorithm is looking for, and so we're partly creating the situation by rewarding basic questions like this. Of course, the SO crowd also fuels the problem by creating positive feedback effects that push such basic questions to the top of the list and keep them there.
I will note that if we weight by the number of incoming questions and remove SO entirely, these numbers are actually more reasonable. Of all non-SO questions coming in, Math seems to get somewhere between 20 and 25% of them right now. That means that on average, a question here is less likely to make it on the list than network-wide. However, the average question here is more likely to top the list than network-wide. This may partly be due to the penalty on multiple questions from the same site. The "hotness score" of the question is reduced by 2% for each question above it in the list from the same site, so that a single site will never dominate the list too much. It could also be interpreted as evidence that users primarily from other sites are at least partly driving these very hot questions.
I'll also note that it's not yet clear how much the new format of the list will change these numbers. Most of my data is from when the list was still in the drop-down menu at the top of all pages in the network. Now it appears (truncated) without any clicks required on the main page of all network sites, and heavily truncated to 3 questions on all post pages as well. This may change how frequently MSE appears and what sorts of questions appear.
What has certainly changed is the visibility of this list. Before, it was quite easy to ignore if you're the kind of user who isn't interested in these things (like most MSE regulars). Now, it sticks out like a sore thumb on every page, so that it's much harder to ignore.