  We show that if $T$ is any of four semigroups of two elements that are not
groups, there exists a finite dimensional associative $T$-graded algebra over a
field of characteristic $0$ such that the codimensions of its graded polynomial
identities have a non-integer exponent of growth. In particular, we provide an
example of a finite dimensional graded-simple semigroup graded algebra over an
algebraically closed field of characteristic $0$ with a non-integer graded
PI-exponent, which is strictly less than the dimension of the algebra. However,
if $T$ is a left or right zero band and the $T$-graded algebra is unital, or
$T$ is a cancellative semigroup, then the $T$-graded algebra satisfies the
graded analog of Amitsur's conjecture, i.e. there exists an integer graded
PI-exponent. Moreover, in the first case it turns out that the ordinary and the
graded PI-exponents coincide. In addition, we consider related problems on the
structure of semigroup graded algebras.
