Note that for cylindrical equal-area projections, the projection cylinder only intersects the globe at the standard parallel if both are at the equator. If the standard parallel is at latitude φ, then the radius of the projection cylinder will be cos²(φ).

There are two factors at play here. Decreasing the radius of the projection cylinder shortens the parallels of the resulting map by a factor of cos(φ). At the same time, the curvature of the globe naturally distorts scale along the meridians, also by a factor of cos(φ). The reference parallel will be located where east-west and north-south distortion are the same. Simple algebra leads to a radius of cos²(φ).

Many thanks — und vielen Dank — to Hans Walser for pointing this out to me.