2 corrected typo

There are a few papers out there dealing with a slightly different focus - algebraic K-theory over the "field with one element" $\mathbb{F}_1$. Some of the frameworks for $\mathbb{F}_1$ algebra include semirings, so these paper papers might contain material that covers what you are interested in as well.

The thesis of Nicolai Durov arXiv:0704.2030 describes a setting for algebraic geometry over a class of objects more general than rings. These objects are commutative algebraic theories. Commutative rings, commutative semirings and commutative monoids all form full subcategories of commutative algebraic theories. Among the many things Durov does in his thesis, he includes some discussion of algebraic K-theory in the final chapter.

The "blueprints" of Oliver Lorscheid (arxiv.org/1103.1745 and its sequels) also contain semirings as a full subcategory. According to the abstract, he will eventually get to K-theory of blueprints, which will contain K-theory of semirings as a special case.

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There are a few papers out there dealing with a slightly different focus - algebraic K-theory over the "field with one element" $\mathbb{F}_1$. Some of the frameworks for $\mathbb{F}_1$ algebra include semirings, so these paper might contain material that covers what you are interested in as well.

The thesis of Nicolai Durov arXiv:0704.2030 describes a setting for algebraic geometry over a class of objects more general than rings. These objects are commutative algebraic theories. Commutative rings, commutative semirings and commutative monoids all form full subcategories of commutative algebraic theories. Among the many things Durov does in his thesis, he includes some discussion of algebraic K-theory in the final chapter.

The "blueprints" of Oliver Lorscheid (arxiv.org/1103.1745 and its sequels) also contain semirings as a full subcategory. According to the abstract, he will eventually get to K-theory of blueprints, which will contain K-theory of semirings as a special case.