I did this analysis to calculate how much of Seismic noise couples to the cavity resonance frequency due to the birefringence of the mirror.
The seismic noise can twist the cavity if the support points are not exactly symmetric which is highly possible. The twist in the cavity will change the relative angle between the fast axes of the mirrors (which should be close to 90 degrees normally). This twist changes the resonant frequency of the cavity as the phase shift due to the mirrors fluctuate.
Edit Tue May 5 11:09:53 2020 :
I added an estimate of this coupling by using some numbers from Cole et al . "Tenfold reduction of Brownian noise in high-reflectivity optical coatings" Supplementary Information. The worst worst-case scenario gives a vertical seismic acceleration coupling to cavity strain of 5x10-13 s2/m (when the end mirros are at near 90 degrees to each other and the supports are misaligned differentially to cause normal force misalignment of 5 degrees). For comparison, the seismic coupling to cavity longitudinal strain is 6x10-12 (from Tara's thesis). Note, that Tara took into account common-mode rejection of this coupling between the two cavities while in my estimate, I didn't assume that. So it is truly the worst worst worst-case scenario and even then an order of magnitude less than the usual seismic coupling we use in our noise budget calculations where Seismic noise is not dominating the experiment anywhere.
So the conclusion is that this effect is negligible in comparison to the seismic coupling through bending of the cavity.