  Let $(X,d,f)$ be a dynamical system, where $(X,d)$ is a compact metric space
and $f:X\rightarrow X$ is a continuous map. Using the concepts of
\textit{g-almost product property} and \textit{uniform separation property}
introduced by Pfister and Sullivan in \cite{Pfister2007}, we give a variational
principle for certain non-compact with relation the asymptotically additive
topological pressure. We also study the set of points that are irregular for an
collection finite or infinite of asymptotically additive sequences and we show
that carried the full asymptotically additive topological pressure. These
results are suitable for systems such as mixing shifts of finite type,
$\beta$-shifts, repellers and uniformly hyperbolic diffeomorphisms.
