There are two test methods described in Aronson's book; White's Reality Check and a Permutation test. At the heart of both is the idea of a "position vector," e.g. a numerical vector of a series of 1, -1 or 0, which correspond to long, short or neutral positions. For example, a vector of
[ 1 1 1 1 1 0 0 0 -1 -1 -1 -1 ]
would represent being long for 5 days, out of the market for 3 and finally short for 4 days. This can be directly applied to a pairs trade such as long stock A, short stock B for 5 days, no position for 3 days and finally short A and long B for 3 days. The Monte Carlo aspect of the tests in question is essentially an n number of random permutations of this position vector.
The difference between the two tests is how the null hypothesis sampling distribution is defined and created. For the simpler of the two, the permutation test, the null is that the "rule" has no predictive power and so the randomized position vector is multiplied with the "returns" to give a distribution of "no predictive power returns." In the book log returns are used, but any return can be used; e.g. dollars made per day on a minimum sized pairs position, the tick value of the spread curve etc. This will simply be the test statistic used for comparative purposes, and the test will be comparing apples to apples.
For White's Reality Check, the null is that the "rule's return" is zero, and so the return vector must be detrended such that a continuous long or short position would give a zero return over the test period. The book subtracts the average daily log return from each daily log return to acheive this because log return is the chosen test statistic. However, if another test statistic is chosen, it too must be detrended in an appropriate way, e.g. subtract the average dollars made per day on a minimum sized pairs position from each individual daily dollar return on the same sized position.
It would, therefore, seem to be quite straightforward to apply the standard tests from the book:
i) create your position vector
ii) create your chosen test statistic return vector (detrended if necessary)
iii) apply the test
However, having written all this, I think the more pertinent problem for pairs trading is data mining bias, wherein the search process for stock A and stock B should be subject to testing, rather than testing A with B in the above framework after A and B have been selected.
You may find it interesting to browse my data snooping Github, which has code and various downloaded papers related to this general area.