Forgot to mention that it really helps anivia to have the AoE effect of randiuns, if your quick during a Frost-Flash-Bite combo you can use randiuns to maximise the damage of glacial storm, also it allows to to really support in team fights.
I have tried frozen heart, but the lack of health is a problem, but this is just my preference.

The lack of health pots is a problem if your not laning mid, in which case cloth armour is a better choice.
Also forgot to add the crystal septer as an alternate to lichs bane. been working with it for a while, but the lack or MR can be trouble some, if the other team is AP heavy.

This is certainly a different build from the norm, but I don't like Randuin's on Anivia. If you're building it just for the CDR and defense on it, you're better off getting Frozen Heart, because it's got even more CDR, armor, and mana which is something Anivia desperately needs more of IMO.

Though... I wonder how you can start off with Ruby Crystal? It just seems very difficult because you don't have any health potions to recover lost HP.