How much topsoil to level a lawn?

The depth cap is the famous constraint. The one that actually stops you is the calendar.

Two university rate tables land on the volume above. Cornell CALS lists a quarter inch as 0.8 cubic yards per 1,000 sq ft, and the LSU AgCenter puts a one-eighth to one-quarter inch dressing at "10 to 21 cu. ft./1000 sq. ft." This page returns 20.8 cubic feet, or 0.77 of a yard, so you can hand either figure to the yard and get the same load.

What no rate table tells you is how long the job runs. LSU's cadence for repeated dressings is "1/8-inch dressing every 2 to 4 weeks", and Cornell treats topdressing as something done "1-2 times per year when the turf is actively growing". Put a genuine hollow through that arithmetic: 2 inches at a quarter inch a pass is eight passes, which even at the aggressive fortnightly cadence is four months, and it cannot be bought back by going deeper. LSU adds "avoid dressing when turf is under stress", which takes the hot weeks out of the middle of that window.

So this page answers a buying question and a scheduling question at once. Up to about an inch, topdress. Deeper than that and lifting the turf, filling underneath and laying it back is the faster repair, not the drastic one.

Then there is what you spread, which constrains you more than how much. LSU warns that "soil layering can affect water movement and root health", and that a half-inch dressing risks exactly that "unless the material dressed was similar in texture to the current root zone". Cornell caps compost at "no more than 1/4 inch per application". Match the texture already under the grass, or eight patient passes will build a perched water table one lift at a time.