Why is aluminium carbonate unstable?
Aluminium carbonate reacts easily with water/water vapour to form aluminium hydroxide as you can see on the Wikipedia page.

$$\ce{Al2(CO3)3 + 3 H2O → 2 Al(OH)3 + 3 CO2}$$

So, if you want to store aluminium carbonate, you have to make sure it is free from moisture.



Why it is unstable:

Smaller and more charged metal cations (i.e. metal cations with high charge density) tend to distort the electron cloud of the carbonate ions (polarisation).

Polarisation eventually leads to abstraction of oxygen from the carbonate ion, making one of the $\ce{C-O}$ bond in the carbonate weak, which makes it very prone to attack by water and heat (similar to the decomposition of copper (II) carbonate when compared to alkali metal carbonates).

The reaction itself doesn't release a lot of energy; hence why it is rather unstable and not reactive in chemistry terms.