If I understand correctly, neither heat dissipation nor existing manufacturing techniques are amenable to this approach. Also modern CPUs do have more than a dozen layers IIRC.
I think op is talking about 6 flat normal chips as the sides. This would allow for cooling stuff inside the cube. Maybe having only 5 of the sides as chips would make it even easier to have a heat sink. The center of the cube could be copper or something.
You can't cool from the inside of that cube without having some way of transporting the heat out of it.
All you'd end up doing is heating that inside up to the temperature of the dies and after that there would be no more cooling effect (and this would happen in a few seconds after starting the whole thing up). You could do an 'inverse' of this by cooling the dies from the outside and having the interconnects in the space in between. This would still require a lot of cooling and there would be an issue with connecting the resulting assembly to the underlying PCB.
I think woah means that the top would be open. So instead of a cube it'd be an open box. A heat sink (or arrangement of peltier coolers attached to a fan and heat sink, or whatever) would fit down into the box and be in contact with the five core-containing sides.
Electronic cooling? Do you mean Peltier elements? They have a 'hot' and a 'cold' side, and their capacity is handily outstripped by any modern CPU. If that worked we would be using Peltier coolers to make quiet PCs today, after all, that problem is only 1/5th as complex.
Another problem is that companies (though not necessarily users) love thinner and thinner phones and laptops, and a cubic CPU wouldn't fit in such a device.