The logic of one-sorted GPA is axiomatized by G.Japaridze and is called GLP. It is a propositional logic with the modalities < n> denoting n-consistency, for all natural numbers n. See

K. Ignatiev. On strong provability predicates and the associated modal logics. Journal of Symbolic Logic 58 (1993), 249-90.

The propositional logic of the many-sorted GPA is naturally
formulated in the language with sorted propositional variables
p_{i}^{n}, where the upper index n
indicates that the variable
ranges over sort n, that is, over arithmetical Pi_{n+1}-sentences. The
assignment of sorts can be extended to arbitrary polymodal
formulas in a natural way (all formulas of the form <n> A
have sort n). In addition to the axioms of the logic GLP,
we have an identity expressing the principle of
Sigma_{n+1}-completeness:

\neg p_{i}^{n} -> [n]\neg p_{i}^{n}.

We should also keep in mind that the rule of substitution of the logic in question is restricted to respect the sorts. Then the above principle in particular allows to derive the axiom < n >A -> [n+1]< n >A of GLP.

The fact that this provides a complete axioamtization should be extractable from the work of Japaridze and Ignatiev.