Quite a bit has been said in regards to design, in particular, whether it is good design or poor design to give a Frost Giant a frost weapon. I'd like to respond to this question purly from a design viewpoint.
Firstly, and arguably most importantly, regular Frost Giants (MM, pg. 122) are NOT given any frost weapons, they are simply supplied with a Huge Greataxe.
Secondly, I have always been under the impression that the possesions of any given creature are suggested possesions and are, by no means, "carved in stone". Therefore, wouldn't it make sense that if, for whatever reason, the Frost Giants in your campaign would not carry such a weapon, you would simply supply them with another equal weapon of your choosing?
Thirdly, the only Frost Giants that are given a frost weapon are the Frost Giant Jarls (+2 Frost Greataxe). From a design point of view, giving a Frost Giant Jarl anything other than a frost weapon would prompt loads of questions. For instance, if you produced a Frost Giant Jarl that weilded a +2 Sonic Greataxe, imagine the feedback you would recieve: "why does the Frost Giant have a Sonic Greataxe? Is there some sort of section of thier culture that I am missing, what is the relavance?"
Fourthly, giving a Frost Giant Jarl a +2 Flaming Greataxe (which would benifit it when used against creatures in its own climate) would produce the same reaction as, say, giving a Fey a Cold Iron Longsword, or a demon a Good Aligned Mace (demons fight primarly with devils, don't they? Wouldn't this make a lot of sense?). Would it seem to you a good idea, from a design standpoint, to equip a demon with a Good Aligned Mace, or would that seem contradictory to the demon's nature?
Finally, check out the section "Frost Giant Characters" (MM, pg. 122). You'll note that Frost Giant clerics' suggested domains are: "chaos, distruction, evil, or war". There is no mention of the Cold Domain. This seems to lend to the idea that Frost Giants are not to be treated soley as cold-based casters. From a design standpoint, this seems to round the Frost Giant out fairly nicely to me.
I would say, with all of these points considered, that the designers of the Frost Giant did an excellent job. Most importantly to your point: don't like that +2 Frost Greataxe? Swap it out with something else. D&D and Wizards of the Coast seem far from "hard and fast" on these rules to me.
Hope this helps,
T from Three Haligonians