  Let $F$ be a fixed field of characteristic zero containing an element $i$
such that $i^2 = -1$. In this paper we consider finite dimensional
superalgebras over $F$ endowed with a pseudoautomorphism $p$ and we investigate
the asymptotic behaviour of the corresponding sequence of $p$-codimensions
$c_n^p(A),$ $n=1,2, \ldots$. First we give a positive answer to a conjecture of
Amitsur in this setting: the $p$-exponent $\exp^p(A) = \lim_{n \rightarrow
\infty} \sqrt[n]{c_n^p(A)} $ always exists and it is an integer. In the final
part we characterize the algebras whose exponential growth is bounded by $2$.
