Does Liandry's % dmg double on attack speed slows?

no.

movement and attack speed are different mechanics.

it only doubles on movement restrictions.

This bonus damage is doubled against movement-impaired units

SOURCE: http://leagueoflegends.wikia.com/wiki/Liandry's_Torment

Patch 3.13 introduced a change to attack speed slows causing them to be treated the same as movement speed slows. Unless there was some kind of undocumented, ninja nerf within the last 5 months (when I last checked this), attack speed slows should increase the damage on Liandry's passive.

Try this out by burning someone from full health with/without a Frozen Heart in your inventory.