Various classical results suggest that stable cohomotopy may usefully be regarded as being the algebraic K-theory over the "field with one element" $\mathbb{F}_1$:

$$ K \mathbb{F}_1 \;\simeq\; \mathbb{S} $$

A summary table with pointers is on the nLab here.

Guillot has made this observation the starting point in his study of "Adams operations in cohomotopy" (arXiv:math/0612327), on which Jack Morava then based his note "Power operations and Absolute Geometry" (2012, pdf). Both of these seem to remain unpublished, though.

Has anyone picked this up? Are there any further developments along these lines?