How much roughly would it cost?
Keep in mind this is just a rough idea, lumber prices change almost daily (it's a fluctuating commodity, like food).
To build a room 12' wide x 8' deep x 8' high, and assuming there is already a floor to build this on, you will need:
A secondary floor - (3) 7/16" thick minimum sheets of plywood. The framing for each sheet will require (6) 8' long 2x4's. That is (18) 2x4's for the floor.
Walls - (20) 1/4" thick sheets of plywood. 10 for the inside of the wall, 10 for the outside. The framing requirements are the same as the floor, so (6) per panel, or (60) total.
You want a thicker plywood for the floor because it needs to support you and your drums. The thicker the plywood, the less flex it will have. If you use too thin of a piece, your floor will sag somewhat. 7/16 should be minimum, anything thicker would be even better.
The walls don't have to be as thick, because they don't support anything, the framing does. The plywood just holds the sound deadening material inside the wall. Looking at my local Lowes site, 1/4" is cheaper than 1/8", so that's why I listed 1/4". Keep in mind that the thinner the wood, the less sound deadening it will do.
The framing needs (6) 2x4's because each piece of plywood needs framing around its perimeter, and studs every 16" on center. That is (5) 2x4's right there. The sixth is for bracing between the studs.
Looking at my local Lowes site, the prices are as follows:
7/16"x4'x8' plywood per sheet: $9.15 (x 3 = $27.45)
1/4"x4'x8' plywood per sheet: $11.65 (x 20 = $233.00)
2"x4"x8' framing per piece: $2.67 (x 78 = $208.26)
Your total in lumber is $468.71 before tax.
If I did this, I would try to fill the walls and floor with whatever kind of foam type material I could find for cheap. Carpet padding comes to mind, some carpet stores will allow you to dumpster dive and get all the carpet and padding you can haul. I would stuff the walls and floor with padding, and hang carpet on both sides of the walls and carpet the floor also.
If you have the tools and the know how, you could build this for less than $750. Don't forget you will need nails, but screws will be better because they don't pull out, and you will need a door of some kind. It wouldn't hurt to buy maybe 10-15 extra 2x4's just to be on the safe side (mistakes will be made, its almost guaranteed). Couldn't tell you about cooling and ventilation, other than try to find a cheap window unit.
This plan is without a ceiling. If you put a ceiling on it you can just use (3) pieces of plywood, unless you want to frame them also. Again, this is a very basic framed room. Hope it helps. It isn't cheap, but it isn't nearly as ridiculous as others have suggested.