It seems commonplace to incorporate ab work into your cool down; probably because at the end of a grueling workout it is so nice to lie down. This is assuming you do both a cool down and ab work, but I have faith that you do both.
Here's an idea though, start your workout with abs and core exercises. After warming up, warm up your center with some ab work and feel how this improves your other activities. When you work out, either lifting or doing cardio, you will be better able to stabilize your torso. Feeling connected to your core will not only help your overall form but it will also help protect your back and prevent injury. For instance, engaging your core will help you take that backward sway out of your bicep curls.
When I start my workout with ab exercises, I feel the heat of my center radiating to my limbs and I feel stronger and more efficient. Try it next time you work out and let me know how it goes.