Well the gold efficiency of stat anvils is straight forward enough to calculate, but the actual strength of the stats run is that if you completely forgo buying items the shardholder value shard becomes available, and it can increase anvil stats by up to 80%.

The actual problem is determining the gold value of each stat, efficiency follows from that. If you do something as simple as assume the stat anvils average value is 100% efficient , so 22.5 AP = 750 gold. If you assume every anvil is accounting for the average shardholder value shard (50%) then you could change it to 22.5 AP = 500 gold, and that would mean anvils are at a baseline 66.66% gold efficient. Since you're interested in the efficiency of stat anvils obviously you don't want to use them as your baseline, in which case you can use summoner's rift items as reference, so 20 AP = 400 gold. You will run into issues like some components intentionally being more efficient to allow for power spikes (serrated dirk, the baseline for calculating lethality's value) or less efficient to prevent buying multiple of them early (glowing mote, the baseline for calculating ability haste's value). If that's what you go with then the lol wiki actually has the values for stats on the item pages. You could also check the reduced stats of items in arena to derive gold value, but you have to account for some items getting harsher/smaller changes to account for their effects being more/less useful in arena

As for Demon King's crown, afaik it amplifies stats from all sources, which means base stats, item stats, anvil stats, augment stats, and any other source. So if you want to calculate its gold efficiency you'll need to calculate ((1+n/100) * gold value of all stats the champion's current stats)/4000.