Does work behave the same way for all temperature range while in a refrigeration setup. When quality of heat changes with the temperature, Does it apply to work also. IF we consider refrigerator Source-sink of 1000-500k and 300-250k, Does 1KJ of work input have the same effect or the effect is different. My doubt may be a very silly, i might be missing something. Any input is appreciated. Thank you.
In theoretical, ideal setup, work will be approximately the same for the same temperature deltas between original temperature of the body, target temperature, and heat sink temperature. Cooling something from 200K to 100K given 300K heatsink should take the same work as cooling it from 1000K to 900K given 1100K heatsink.
In reality, obtaining efficient heat sinks for low temperatures is difficult. You can just leave something sit in open air for a couple minutes to bring it from 1000K to 900K while you need to jump through hoops and spend a lot of energy to get something from 200 to 100K - simply because your heatsink will be at ~300K in both cases. If you can provide a heat sink of temperature that is the same offset from your target temperature, you have comparable work - but usually, you can't.
There are also operational temperatures of the coolant. Most heat is stored/extracted at phase change, gas<->liquid. This needs to correspond to boiling point of the coolant, so you'll use e.g. sodium to cool a nuclear reactor, but helium to cool superconducting magnets. Since they have different enthalpy of evaporation, they will provide different performance - not due to difference in temperature, but due to being different substances, that happen to boil at different temperatures needed for their respective processes - affecting performance of the refrigerators.