  Superstatistics describes statistical systems that behave like superpositions
of different inverse temperatures $\beta$, so that the probability distribution
is $p(\epsilon_i) \propto \int_{0}^{\infty} f(\beta) e^{-\beta
\epsilon_i}d\beta$, where the `kernel' $f(\beta)$ is nonnegative and normalized
($\int f(\beta)d \beta =1$). We discuss the relation between this distribution
and the generalized entropic form $S=\sum_i s(p_i)$. The first three
Shannon-Khinchin axioms are assumed to hold. It then turns out that for a given
distribution there are two different ways to construct the entropy. One
approach uses escort probabilities and the other does not; the question of
which to use must be decided empirically. The two approaches are related by a
duality. The thermodynamic properties of the system can be quite different for
the two approaches. In that connection we present the transformation laws for
the superstatistical distributions under macroscopic state changes. The
transformation group is the Euclidean group in one dimension.
