  The thermal-to-percolative crossover exponent \phi, well-known for
ferromagnetic systems, is studied extensively for Edwards-Anderson spin
glasses. The scaling of defect energies are determined at the bond percolation
threshold p_c, using an improved algorithm. Simulations extend to system sizes
above N=10^8 in dimensions d=2,...,7. The results can be related to the
behavior of the transition temperature T_g (p-p_c)^\phi between the
paramagnetic and the glassy regime for p-> p_c. In three dimensions, where our
simulations predict \phi=1.127(5), this scaling form for T_g provides a rare
experimental test of predictions arising from the equilibrium theory of
low-temperature spin glasses. For dimension near and above the upper critical
dimension, the results provide a new challenge to reconcile mean-field theory
with finite-dimensional properties.
