Stability of the entropy equation

In this paper we prove that the so–called entropy equation, i.e., $$H(x,y,z)=H(x+y,0,z)+H(x,y,0)$$ is stable in the sense of Hyers and Ulam on the positive cone of ${\mathbb{R}}^3$, assuming that the function $H$ is approximatively symmetric in each variable and approximatively homogeneous of degree $\alpha$, where $\alpha$ is an arbitrarily fixed real number.