I've kept wild caught bimac octopus for about four years, at about 60 degrees. I decided not to use live rock because I thought it would look wrong to have tropical live rock in a biotope tank with only local So Cal animals. I also wanted to reduce the chances of exposing my bimac to any non-native parasites or pathogens (and also because I'm to cheap to pay for live rock). I use 9.5 gallons of bio-balls in a wet/dry trickle filter, and a deep sand bed in a 60 gallon tank. I get slow nitrate buildup when I keep just an octopus and feed sparingly (the DSB can handle it) and fast nitrate build up when the tank is full of other animals (sea stars, gorgonians, strawberry anemone) and I feed heavily. It took about six months (at 65 degrees) before the DSB (anaerobic) was able to keep the nitrates down, but the aerobic bacteria in the bio balls was ready after about three months. Then I dropped the temp down to 60 (55 for about 6 months).
The bio-filtration is much slower in cold water, but so is the metabolism of the animals, so you can feed less often to partially compensate.
One advantage of the trickle filter is that it really oxygenates the water well, which is good for an octopus, since they are sensitive to low oxygen, and since I covered the tank completely to insulate, and to prevent escape. Good mechanical filtration is a must with bio-balls, and a good idea with a cold tank in general, because most cold animals can't use light to make food, and so must be fed a lot, and you want to get the uneaten food out of the water fast.
FWIW. I found that the 6 to 1 ratio of water to bio-balls works great (less may also work great), and the in-tank DSB does reduce nitrates, but not enough to keep up with the heavy feeding of a fully stocked tank. For my next cold tank build, I'll still use bio balls, filter floss, and a skimmer, but I'll add:
1) An alcohol-fed nitrate filter (home built, with an automatic dosing pump)
2) A CO2 scrubber (easy home build) to reduce the CO2 lever of the air going to the skimmer and keep my PH from falling (a problem in cold tanks)
3) And a poly-pad (to remove copper, and other contaminants chemically)
With that set up I will be able to load it up with animals and feed it an appalling amount every day.
It's not too hard to catch your own bimac if you know where, when, and how to look, and can get to the beach in So Cal (and yes, it's legal with a fishing license to catch octopus). PM me if you want to know more about the when where and how.