Does acceleration increase as gravitational potential energy increases?

Gravitational potential energy can be defined as PE = mgh, where m is mass, g is the acceleration due to gravity, and h is the height above the surface of the earth. By its very definition, the gravitational acceleration cannot change, but in reality it's a little more complicated.

First, consider that the potential energy as defined in the equation above will increase if either the mass or the height of the object increases, but we cannot alter the value of g, which is just a specific value for acceleration. This is because the force being generated by gravity is a fixed value, and the mass of the object and its height are irrelevant because they're not the factors generating the force. 

However, gravity is an inverse-square force, meaning that it gets exponentially weaker as one gets farther away from its source. This means that something experiencing the earth's gravitational force is going to feel a different degree of acceleration based primarily on its distance from the origin. For example, if astronauts in the space station were experiencing 9.8m/s2 of gravitational acceleration toward the earth, then they wouldn't appear to be floating; they'd look just like they do on the surface of the planet. 

Thus, we can say that on a large enough scale, acceleration will change depending upon height. However, this doesn't take potential energy into account; if we were still using the simplified equation given above, it would appear that acceleration decreases as potential energy increases, which doesn't make sense.

The more complete form of the potential energy equation is very simple to Newton's Gravitational equation; PE = GMm/r. That is to say, potential energy equals the products of the gravitational constant and the masses of the two bodies, divided by the distance between them. From this, you can see that the potential energy will decrease as the value of r increases, which explains why the astronauts actually have very little potential energy.

So, to answer the question directly, we can see that gravitational potential energy will increase as one gets closer to the gravitational origin. Acceleration will also increase, but this is because the gravitational force is increasing as well. Thus we need to acknowledge that acceleration and potential energy change proportionally within a gravitational field, but that this is not a causality relationship, and it doesn't work if you're using the simplified PE = mgh equation.