Here's the thing, an ice cube does not "give out" coldness. There's actually no such thing as coldness. What it really does is absorb heat that's already there. The laws of thermodynamics tell us that a substance at a higher temperature will gradually lose heat to a substance at a lower temperature. Since ice cubes are usually put into a drink when they are about 0 degrees C, the liquid they are suspended in will lose heat to them. What this does is raise the temperature of the ice cube and lower the temperature of the surrounding liquid. The result is a temperature somewhere in between but one in which the heat is more evenly distributed.
What we can now do is a neat little trick using the zeroth law of thermodynamics which is to remove the liquid from the equation. If A (the air) can exchange heat with B (the liquid) and B can exchange it with C (the ice cube) then A is effecitvely exchanging heat with C. So what we're talking about is an ice cube in summer compared to one in winter. Now the question becomes a little clearer.
Picture an ice cube sitting on a cold plate compared to a hot one. The hot one will melt first because heat is more readily transferred than in the cold system. But the cold temperature and the hot temperature will go down by the same temperature. Here's what I mean: 20 degree day vs a 0 degree ice cube might reach a thermal equilibrium of let's say 10 degrees. That's a 10 degree change. A 30 degree day vs a 0 degree ice cube of the same size will reach a temperature equilibrium of, say, 20 degrees. In other words, both systems have cooled down by the same amount. The hotter one just did it quicker.
But then the hotter one starts absorbing more heat from the air, so the drink will also "warm up" faster. The cold one will end up cooling down much later, but it will take a lot longer to "warm up" again. Ultimately, in both systems the final result will be identical. An equilibrium will be reached between drink, water and air. If you keep everything the same size, then the amount of difference should be identical. So, by my back-of-envelope calculations, it makes no difference...I think.