In addition to Barrat_sab, we could carry out a very rough estimate on heat loss from a standard National broodbox over winter.
Lets say the cluster is in the top half of the box, so most conductive losses will be shared by the crownboard and the sides (about an equal area for each?) so lets be really rough here and assume energy loss is 50% through the crownboard (it will be far more than that as the side walls are 3 times thicker for the most part). Reducing the roof losses by a factor of ten will make them almost negligible. That will of course mean slightly more heat loss through the sides, but small enough to be negligible, too. No computer modelling here!
So we may have nearly halved the heat losses through the top half of the hive. Now let's be really assumptive here and say 50% of overall hive losses were through ventilation and the bottom of the hive. So now the savings are 25% or the bees' heating costs for the winter. Assume 10kg of sugar is ample for thermal energy requirement and we have an instant (very conservative) saving, on cost, of 2.5kg of sugar. That capital expenditure has a 100% pay-back in less than one winter (even at one pound fifty for a single sheet of insulation).
Now consider the advantages to the bees; less stores to consume for that energy balance, less build-up of materials of defaecation within the bees (so longer between cleansing flights), less likely to use all stores in one area and need to move across the hive to other stores (potentially a colony threatening activity if the need occurs during a very cold spell). Added to that earlier brooding is likely and as most stores (I would say in this scenario) are required for feeding the early brood, the colony is well placed for a good start in the Spring, with less chance of failure or reduced activity due to stores levels.
All in all, a 'win-win' situation for the minimal outlay for a single sheet of expanded polystyrene. Further to that, I think my workings give a very conservative value for energy savings associated with the energy losses from the crownboard.
Regards, RAB