  We investigate the low-temperature critical behavior of the three dimensional
random-field Ising ferromagnet. By a scaling analysis we find that in the limit
of temperature $T \to 0$ the usual scaling relations have to be modified as far
as the exponent $\alpha$ of the specific heat is concerned. At zero
temperature, the Rushbrooke equation is modified to $\alpha + 2 \beta + \gamma
= 1$, an equation which we expect to be valid also for other systems with
similar critical behavior. We test the scaling theory numerically for the three
dimensional random field Ising system with Gaussian probability distribution of
the random fields by a combination of calculations of exact ground states with
an integer optimization algorithm and Monte Carlo methods. By a finite size
scaling analysis we calculate the critical exponents $\nu \approx 1.0$, $\beta
\approx 0.05$, $\bar{\gamma} \approx 2.9$ $\gamma \approx 1.5$ and $\alpha
\approx -0.55$.
