Standard deviation of spectral flatness -- so what am I measuring, conceptually?
Since you are interested in the "flatness" of your spectrum, in fact, you are interested in how close your signal is to a white noise (that by definition has a flat spectrum + random phases). If you step back, one measure would be the "distance" of your observation to the white noise reference.

The obvious measure in terms of information theory is Kullback-Leibler divergence. You do not need to understand every part of it, but it measures in bits (if you use log base 2) the distance between both distributions.

The good thing in your case is that your reference is flat, so that what remains is the entropy of your spectrum. There are many existing implementation (for instance in scipy).

Note that you are still on the safe side: if your distribution is approximately gaussian, both measures (entropy and std) will be proportional. The entropy is however more general and more principled. As an extension, you will be able to generalize to other types of noises (1/f for instance).