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The free energy and the specific heat of the two-dimensional Gaussian randombond Ising model on a square lattice are found with high accuracy using graphexpansion method. At low temperatures the specific heat reveals azero-temperature criticality described by the power law $C\proptoT^{1+\alpha}$, with $\alpha= 0.55(8)$. Interpretation of the free energy interms of independent two-level excitations gives the density of states, thatfollows a novel power law $\rho(\epsilon)\propto \epsilon^\alpha$ at lowenergies. An exact high-temperature series for this model up to the term$\beta^{29}$ is found. A proof that the density of one-site spin flip statesvanishes at low energy is given.

Zero-temperature criticality in the Gaussian random bond Ising model on a square lattice