Is it OK to use regression when analysing a difference of variables? Consider an experiment where we want to check if the change in a variable X between 2 time periods (X1 at t=1, X2 at t=2) is caused by independent variable A or B (which do not change over time).
Would the following analysis make sense: a regression where DX = X2 - X1 is the dependent variable and A & B are the independent variables? Or can you not take the regression of a difference of variables?
 A: Yes that could make sense.  But you are making more assumptions that if you treat X1 as a baseline variable and fit a model like Y ~ X1 + B.  This allows the slope of X1 on Y to be something other than 1.0, which happens when you have measurement error or regression to the mean.  More about this is in the transformation and change chapter of BBR.
A: You should look at panel data to think about other possibilities.
If A and B are constants, $\Delta A$ and $\Delta B$ will be zero too, and constants in your model will not help you.
There are methods for analysis of time change, like first difference estimator or fixed effects, but them will not help you if you hope to find causality in a model made only by constants as independent variables.
First difference estimator will clear your mind:

The estimator is obtained by running a pooled OLS estimation for a
  regression of $\Delta y_{it}$ on $\Delta x_{it}$ [...]

Doing that eliminates changes that are constant from any $t-1$ to $t$
In your example, the variables probably will be not even correlated... so you have to model your experiment differently.
