Where is a Hollow Object's Center of Gravity?
Let's imagine that the Earth's moon is hollow and the remaining mass is structurally strong enough to retain its spherical shape (like a really big tennis ball).
Given this scenario, where would its center of gravity be? What would the conditions be like inside? Also would one still be able to stand on the moon?
Amusingly, the astrophysicist Shklovsky suggested in the 1950s that Mars moon Phobos might be a hollow sphere. Indeed, today it is estimated that about 30% of its volume is made up of empty cracks/gaps. https://en.wikipedia.org/wiki/Phobos_(moon)#Shklovsky.27s_.22Hollow_Phobos.22_hypothesis
If by "a hollow object" you mean a spherical shell -- a theoretically perfect sphere Moon, with a perfect sphere of material removed from the center: The center of mass, and therefore the center of gravity remains at the geometric center. Ref Newton's Law; Bodies with spatial extent.
The body of the moon is plastic; Meaning the gravitational forces are strong enough to reshape the material... it basically flows into a sphere. So if you magically removed a small amount of the center, it would collapse into a smaller sphere. If you removed most of the center, then the "thin" shell may or may not be strong enough to support it's shape.
Inside a hollow spherical shell, there is no net gravitational force. So you would be in a zero-G environment everywhere within the shell. From outside the shell, the gravity is the same as if all the remaining mass was located at a point at the geometric center.