What will companies do? I've known companies that wanted pilots to shut an engine down on descent to save fuel. Who wanted pilots to badger ATC constantly for direct, who compared fuel burns on every trip and called pilots on the carpet if theirs was different than others. Who barked about tenths of an hour difference on trips, and who seemed to feel that delays such as weather and ATC holdups were the fault of the pilot (couldn't you just have flown through it? Why didn't you cancel and go in VFR?) and so on. I knew one company VP who insisted that pilots go to minimum necessary power all the time...including takeoff. Don't use any more than absolutely necessary, don't burn any more fuel than absolutely necessary.
On the other hand I came back to the airport early one morning, after sitting at a hospital with a medical crew while they recovered a heart. The Chief Pilot had stayed behind with the airplane, as he was afraid of hospitals...he thought he'd get sick. The weather was clear, but bitter cold. When I climbed into the cockpit, I found a 600lb fuel balance problem, and discovered he'd been running the engines to keep warm. He'd bring one up, heat the cabin with bleeds, and shut it down again. It was a short trip so we had adequate fuel to go home, but he'd burned a lot. He was a shareholder, and could get away with that. He pushed other pilots hard enough that one of them routinely came home with the low fuel light on.
I've never been paid a fuel bonus, though I've saved hundreds of thousands of dollars over the years by judicious fuel management and useage. I've been threatened with my job, however, by owners who thought that somehow that might save them a few dollars, and who didn't seem to appreciate that I was already saving them more than my annual salary. Talk about biting the hand that feeds.