For L × L square lattices with L ≤ 20 the 2D Ising spin glass with +1 and -1 bonds is found to have a strong correlation between the energy and the entropy of its ground states. A fit to the data gives the result that each additional broken bond in the ground state of a particular sample of random bonds increases the ground state degeneracy by approximately a factor of 10/3. For x = 0.5 (where x is the fraction of negative bonds), over this range of L, the characteristic entropy defined by the energy-entropy correlation scales with size as L 1.78(2) . Anomalous scaling is not found for the characteristic energy, which essentially scales as L 2 . When x = 0.25, a crossover to L 2 scaling of the entropy is seen near L = 12. The results found here suggest a natural mechanism for the unusual behavior of the low temperature specific heat of this model, and illustrate the dangers of extrapolating from small L.