OK, I took the advice of everybody here and built out an 8 x 8 box frame using 2 x 6 pressure treated wood. I decided on using 2 x 6s vs 2 x 4s to give me a little more wiggle room for a more solid base. I plan to fill the box with 3-4" of 2a limestone aggregate which is a mixture of fine stone up to 1.5 inches. I've used this stuff for patio base and it packs down almost as tight as concrete.
The concern I had is once I built out the box frame, I noticed that it is not touching the ground on all sides. This is due to the fact that the patio is motared brick and has high and low spots. I wasn't sure if I should be worried about the gravel washing out of the box because of this. The gaps are not very large but they do exist. Should I be concerned or will the gravel stay put once it's packed down inside the box? I was thinking about putting down some type of plastic or landscape fabric and stapling it to the inside of the box to hold the gravel in place. I'm not sure if it's worth going to this trouble or not.
The only other question I had was do you think it's necessary to secure the box frame to the patio? My thought was the the box was heavy enough that it really shouldn't move once in place, but I wasn't sure how much pressure would be exerted on the box once the gravel and tub are sitting in side.