  In this paper, we investigate the asymptotic behavior of optimal designs for
the shape optimization of 2D heat equations in long time horizons. The control
is the shape of the domain on which heat diffuses. The class of 2D admissible
shapes is the one introduced by Sver\'{a}k, of all open subsets of a given
bounded open set, whose complementary sets have a uniformly bounded number of
connected components. Using a $\Gamma$-convergence approach, we establish that
the parabolic optimal designs converge as the length of the time horizon tends
to infinity, in the complementary Hausdorff topology, to an optimal design for
the corresponding stationary elliptic equation.
