How do you compare term counts between two different periods, with different underlying corpus sizes, without bias? I'll set the question up with an example. You are analysing news coverage text data from 2014, and find that a term appears less often in the third quarter of 2014 than the final quarter (let's imagine it's the term "Christmas"). Unfortunately, there are also far less news articles in the third quarter than in the second (due to the lack of news in the summer). So how do we accurately compare the counts in each quarter? We assume that there will be a greater number of occurrences in the fourth quarter, but how much does the magnitude of this difference depend on the change in size of the underlying text?

Heap's law shows the relationship between text size and number of unique terms. It's non-linearity implies that the rate of new, unique words introduced by the text decreases as you increase the size of the text, and the proportion of the text taken up by each existing word therefore increases. This applies given documents taken from the same 'distribution' of text, in other words the underlying zipfian distribution of word ranks is identical (see wiki). 

In my example above this is obviously not the case, since the underlying topics, and resultant term distribution, will change between summer and winter, especially with regards to the term "Christmas". But take the same term count but over the whole of 2013 and 2014; you would reasonably expect the general underlying term distribution to be the same in each period, so Heap's law applies, but what if the volume of text has changed? Simply normalising by the size of the text, or the number of documents, does not, as far as I can tell, account for the relative change in expected value of the term count.

I have a hunch that this might be a simple application of Heap's or Zipf's laws, but I can't see how to apply them to this particular question. Appreciate any insight.
The approach suggested by j.a.gartner is good if you want to analyze the quarter-to-quarter change in frequency for a given term, considering only that term in isolation.  What it doesn't do is evaluate the relative frequency compared to all other terms in the corpus.  For this you could compare frequency rank for all terms quarter-to-quarter. This is the analysis used in Zipf's Law.  Another advantage of this analysis is that you can test whether the generating process is or is not stationary (e.g. governed by same distribution quarter to quarter).

The advantage of comparing frequency rank is that it doesn't depend on the relative size of the corpus for each quarter (as long as they are all "large").  In simple language, the frequency rank of the word "Christmas" in the 4th quarter will probably be the same, regardless of whether the (US English news) corpus is 10,000 articles or 1,000,000 articles.