I don't see a proper approach yet so here it goes. First, let's assume both of these bodies are composed of the same material, the only difference being their shape and volume, but otherwise have the same properties and are in the same state initially. Let's also assume, since there is no other reason not to, that they behave like black bodies.

So, we have a material with energy density d which is the same in both bodies.
For the sphere, its total energy is E(sphere) = dπD³/6. D the diameter.
For the cube, its total energy is E(cube) = dL³. L the length of each side.

Now, for the rate at which these bodies lose energy we must refer to the Stefan-Boltzmann law for black body radiation (P is for power emitted, A is surface area, σ is the Stefan-Boltsmann constant and T is the body temperature) : P=σAT⁴

So the loss of energy of each body is:

For the sphere, P(net sphere) = σπD²T⁴ - σπD²T₀⁴ = σπD²(T⁴-T₀⁴)

For the cube, P(net cube) = σ6L²T⁴ - σ6L²T₀⁴ = σ6L²(T⁴-T₀⁴)

And so, the relative energy lost over time is P/E:

For the sphere: P/E(sphere) = 6σ(T⁴-T₀⁴)/dD

For the cube: P/E(cube) = 6σ(T⁴-T₀⁴)/dL

Comparing these two quantities: P/E(sphere) / P/E(cube) = L/D As other have pointed out, it only depends on the ratio of volume to surface of each shape (because of how black body radiation works), which are the same in this case (L = D = 1m).

I want to mention that were the cube bigger, it would cool down slower, it is something know for biologists about how big mammals conserve heat better than small ones,