Which cycling backpack stops back sweating

I doubt that you'll find anything.

Why not mount a rear rack and carry your stuff in a pannier bag? That will not only keep your back free, it will also be safer because of the lower centre of gravity.

There are a couple of different ones available which have a mesh stretched over a concave internal frame. The mesh rests against the back providing an air channel between body and the actual pack.
These do improve things, but sweating during exertion is really hard to avoid entirely.
And unless your clothes are well ventilated too odds are your back will get sweaty anyhow.
If you have a look at wind vests for cyclists you'll see that they have front panels in regular cloth while rear panel is mesh spcifically for this purpose.

I'll side with Yojimbo, (unless you're into MTB-ing) rack + pannier is much nicer than carrying a backpack. For lighter loads I'm not too concerned about centre-of-gravity though.

At the risk of seeming terse, get it off your back and on your bike. There are plenty of seatpost mounted bags that will take your stuff, the only limit is size. From things like this or even as big as this Both easily fitted to the bike and in use by commuters I know.

If you're still looking for a backpack, any of the Camelbak NV line will do about as good as you're going to get as far a providing ventilation to the back. As described by dabac, they're designed so that a stretched mesh is the only thing contacting your back and leave around a 1/4 to 1/2" air gap between the pack and your back. If you don't want water, don't put the bladder in.

I bought myself a really cheap rucksack and superglued/stiched a couple of pieces of pipe lagging to it. I stuck on at the top and one at the bottom, both running across the bag. The bag sits about an inch from my back and lets air flow through. It isn't perfect but I bet you'll pay an awful lot of money to get something better. Try it.