So a bunch of the other comments have mentioned this but you would be creating a thermal battery essentially. These can be useful for smoothing out the temperature changes in that room but it isn't exactly efficient since the only way to heat or cool it is by changing the temperature at the surface of the container.

Adding passive heat sinks like radiator fins would increase the efficiency as it would absorb or diffuse the temperature difference with increased surface area but it would still would be subject to things like the air conditioning turning on and off more regularly when there is a higher ambient temperature delta or condensation when the weather is hot and moisture is high. You've essentially added an inactive water boiler tank in the middle of a room that takes up space and takes a long time to either heat up or cool down and it still would be lagging behind where you want the temperature to be.

You're on the right track to a good idea with trying to store thermal energy but it can be made better with a few tweaks:

• Let's make the tank part of an active system by adding pumps and a heat exchanger that integrates with your current air duct system (assuming you have one). We can heat and cool the tank directly instead of passively so that our time and energy is directed more efficiently.
• Insulate it so that we minimize any unwanted heat changes
• Move it to a utility room or outside so you aren't taking up room space

Now we have a thermal battery that works with your air conditioning system as opposed to against it. This can be paired with other methods of heat/cooling such as a solar system.

But if you're in a dorm or somewhere you can't make changes, it could make sense if you aren't paying for electricity, you actively heat/cool the bucket by putting it in a freezer or on a heater/fire, and you don't mind a large metal container in the middle of the room? Just watch for a lot of condensation when cooling the air.