Yes and no.
And now to elaborate...
I have the same card, two actually. THE COOLER IS THE BIGGEST PIECE OF SHIT EVER!!! Friggen I knew it sucked going into it, but for $120/ea card I couldn't pass it up -_- Anyways I've been trying to find a new heatsink, or make these ones better, no luck short of the stupid one AC offers. Out of the box, the bolt pattern on the cards are both a square BUT the distance is different! It's a larger square on the 5800, than it is on the lower end models. That is the "NO" to your question.
Now for the "YES", this is what it'll take... The back side of the AC cooler shows 4 "tabs" that are inside enough to line up on the 5770's smaller square!! What it would require to fit? Well simply drilling out the tabs in the same spot as your holes and then tapping out the holes with threads (6-32 tap, which is 6mm diameter @ x32 threads per inch, which I coincidentally have).
HOWEVER the big difference with most of the 5800 and 5700 is the GPU location! The 5800 is to the slot opening end, where as the 5700 is to the PCIe Power Plug end. Now that doesn't make much of a different if you have a tech-bench case setup, or a super spacious tower case, but what it translates into is the heatsink extended a good amount (probably a whole fan's amount, or more) past the end of the card! Would result in quite a goofy looking card setup lol
I tried contacting MSI in order to do like what I did with my x1950 Pro (Sapphire), which is that I contacted HIS and was able to procure (for the cost of shipping from Asia) the heatsink assembly of the IceQ3 Turbo. Reason I wanted it was the x1600XT I had before was that and it was made by AC! Plus it kept the card cool as hell
Th x1950 one came with memory and VRM sinks to boot! MSI sadly wasn't willing to help out in the same regards
I had hinted slightly that if they wanted to swap me cards, for the HAWK of which I was asking for the heatsink from, I'd be willing to convert solely to MSI from here on out for all my system components lol The guy said the HAWK wasin such high demand that it resulted in high backorders, which I sort of took as a "If it wasn't, then it may have been a possibility" reply
When I specifically asked if he could sell the HSF setup, his reply was that the HAWK setup requires an MSI card's build quality since it is fastened with actual bolts, not plastic retaining pins. I corrected him by letting him know that is how the ASUS is, but he didn't say anything. I think he was trying to get by with that hoping I didn't know better
So that's basically where we sit, except I have 2 furnaces. One that is more akin to a home's heater, where the upper card being in tight confinement is closer to a funeral home's crematory furnace!
I'm thinking about pulling the heatsink off a THIRD time and actually lapping the base to be flat, because there are 2 HUGE milling ridges in it, in a spiral pattern. The first time it was off I polished the base to a mirror finish + new thermal goop (a few minutes with metal polish and a drill acting like high speed buffer), but after just taking the cards out and putting back in where the temps went to over 100F on the top card, I pulled them back off and put fresh thermal goop on again... The temps after the polish and regooping were the same as stock, then after the re-apply after the freaky-temps, they are again back to stock :\ Which only leaves that the ridges are too great to give good enough thermal transfer...
I just hate this CUcore cooler
The card otherwise I'm sure is great! I somehow can get my memory up to 1395Mhz on both, and they use the same memory as most 5870s do which I don't see breaking but to around 1300MHz, so they must have done something right haha It's just there is literally no way the box advert is accurate by saying 11% cooler.
EDIT: JUST noticed something. You are talking about the TWIN and I was thinking about the Tri-Fanned EXTREME model! Appears like in your first picture above, it has threads for what I assume is the holes for our 5770s. I don't know if the Reference board has standard 5800 holes or not though. If you own that board in your picture, measure their distances on the cooler for me!