Yes, the value of acceleration can change. When more gravity pulls on an object it is most likely going to make the object, whether it is a car, or a rocket, to try harder. The value of acceleration will vary most in objects that travel straight up, like rockets. Just like more drag is going to make object try harder to thrust, gravity can most affect the acceleration of vertical traveling objects such as rockets. So in Earth, the gravity is different from many other planets in our solar system. If a spaceship were to take off in Jupiter, it is going to need Newtons more force to even barely get off the ground. Smaller planets that have less gravity, like the moon, will make spaceships try not as hard and possibly use less Newtons of force to take off.
Hope this helps