I avoid hard mode in general because the difficulty is stupid to me. All that really changes is that everyone hates you more, you get more pirates and eco-disasters, Civilization Stage is just other nations advancing faster than you and sending constant streams of war machines, etc.
Buuut if you wanna spit the game's unbalanced difficulty in the face and do it anyway... Well, my advice is to just fight them the way you normally would, because frankly, the Grox themselves aren't that different. Your biggest concern is the aforementioned increased frequency of attacks (but then before going to war with the Grox you should already make sure your planets are well-protected from this), as well as the fact that you have less health.
So just look up general ways of fighting the Grox, if you haven't. Tons of good guides and advice. My personal favourite method (which admittedly isn't optimal) is to raise the T-value of their planets so they all die from too healthy conditions, or create and uplift species and help them form mini-empires amidst the Grox's territory. But that's just because it's fun to screw with them.
Also, I was really happy to scroll down and see such a recent question. For some reason dead communities really get me down, so even though I'm honestly not THAT big a Spore player, I'm always glad to see that the community is still alive.