It would depend on the composition. If Earth's atmosphere were 0.2 bar and if carbon dioxide were still 270ppm, Earth would be 2-8C cooler than it is now. But if Earth's atmosphere were 0.2 bar, but if we did that by removing just nitrogen, then the almost pure oxygen atmosphere would cause just about everything to burn. But if Earth's atmosphere were 0.2 bar, with 20% oxygen and the same partial pressures of carbon dioxide, nitrous oxide and methane, life would have adapted to the lower partial pressure of oxygen. Earth would actually be warmer because the relative humidity would be closer to 100%. Air does not hold moisture. Air pressure actually suppresses evaporation. If you expose a glass of water at room temperature to a vacuum, the water would boil.
A thinner atmosphere would be catastrophic for Earth. Firstly, it wouldn't be thick enough to hold on to the heat from the sun, therefore the Earth would be cooler as a whole.
A thinner atmosphere would also mean less pressure, meaning more of Earth's water would evaporate and more of the world's ice (especially at higher altitudes) would sublimate into space.
Then there's the atmosphere's ability to hold on to oxygen. With a thinner atmosphere, the Earth will only be able to hold on to heavier gases, such as carbon dioxide which would of course suffocate us all. It wouldn't be any better for plant life either, only tundra mosses would be likely to survive. But with a thinner atmosphere the sun's UV would kill off most life.
Any particular reason you are interested in this subject? The atmosphere isn't one particular pressure, so I assume you mean "if the average surface pressure was 0.2 bar (20 kPa, 3 psi) instead of 1 bar (101 kPa, 14.7 psi)".
At this point, I think we would be worried about far too many other things than climate. We would certainly be cooler as some suggest. But more importantly, life would not have advanced beyond single cell organisms, as the UV radiation that would now make it to the surface would have prevented that.
I am assuming that my one thumbs down (so far) is either some one who hates EVERYTHING I say, or a creationist, who doesn't believe in evolution from single cell organisms :^)
At 0.2 bar, a thiner atmosphere would make the earth significantly colder, just have a look at Mount everest which is near the equator, but always covered in ice.
All will be suffocated. Then the Earth will go back to normal..
Lets say the Earths atmosphere were 0.2 bar. How would the climate be affected and how much would daily temperatures be affected?
