The tolerance of rio.warp.reproject(), called error_threshold in GDAL.Warp() or -et in gdalwarp (see https://gdal.org/en/stable/programs/gdalwarp.html#cmdoption-gdalwarp-et) is defined as a constant, preventing users from specifying their value for an approximate transformation:

I came up on this while trying to reconcile differences in a chunked implementation of the reprojection with the reprojection on the full array.
It turns out that the tolerance in the transformation defined on a different chunk can create pretty large differences between the full-array reprojection and chunk reprojection, in addition to the X/Y scaling factors (see #2995).

I'll open a short PR to fix this!