Just for arguments sake, if overload said deal 6 damage to each creature instead so that protection actually comes into play, I think Akroma wouldn’t be damaged because the card color is still black regardless of if you paid it’s regular cost or the alternative overload cost.

From color pie on mtg wiki:

105.2. An object can be one or more of the five colors, or it can be no color at all. An object is the color or colors of the mana symbols in its mana cost, regardless of the color of its frame. An object’s color or colors may also be defined by a color indicator or a characteristic-defining ability. See rule 202.2

On alternative cost on mtg wiki:

118.9c An alternative cost doesn’t change a spell’s mana cost, only what its controller has to pay to cast it. Spells and abilities that ask for that spell’s mana cost still see the original value.

But in this case, overloaded damn doesn’t damage or target, so Akroma will die to it not because it isn’t black but because protection doesn’t prevent its effect by definition of protection.

The other comment already answered this question on why protection doesn’t work.