Until recently the only compressor options were the Engel who do not on sell their compressor and the Danfoss German made compressor that every other brand uses. Of course, there are now Chinese clones.
The latest Engel has by far and away the lowest power consumption. I know from my own experience and tests that Engel uses 50% less power than Danfoss as a fridge and 75% less power as a freezer. I am pretty sure you will find the Chinese clones will use more power than a Danfoss from what I have heard.
The most important parameter though is how much insulation is used. Eg how thick it is as this is where the heat loss (or cold loss) is most influenced.
It all depends on intended use. If you want to sit in one spot and run off solar, power consumption IS important.