  We apply new results on free boundary regularity of one-phase almost
minimizers in periodic media to obtain a quantitative convergence rate for the
shape optimizers of the first Dirichlet eigenvalue in periodic homogenization.
We obtain a linear (with logarithmic factors) convergence rate for the
optimizing eigenvalue. Large scale Lipschitz free boundary regularity of almost
minimizers is used to apply the optimal $L^2$ homogenization theory in
Lipschitz domains of Kenig, Lin and Shen. A key idea, to deal with the hard
constraint on the volume, is a combination of a large scale almost dilation
invariance with a selection principle argument.
