How does the efficiency of the rocket engine using liquid propellants depend on the throttling setting? (Is it more or less efficient while working on eg.: 85% of its maximum thrust?). The rocket engine is in vacuum.
Corroborating Russell Borogove's answer, some Stennis test data I have from 1987 on three different SSMEs shows a small drop in Isp with power level. From 109% to 100% the Isp dropped about 0.08%. I can't find data at lower power levels but my recollection is that the trend continued, with a small degradation in Isp as you throttled down.
For reference, the original SSME could throttle from 109% to 65% of Reference Power Level. A "bi-stable turbopump anomaly" limited the lower end of the throttle range to 67% in the latter part of the program. The throttle range actually experienced in flight was from 65% to 104.5%.
It's surprisingly difficult to find a good answer to this question.
Generally, the rated full power level is where the engine is going to be most efficient.
According to Sutton's "Rocket Propulsion Elements", typical deep-throttling engines suffer between 1.5% and 9% reduction in specific impulse (fuel efficiency) at low power levels. It mentions an outlier, the engine in the Lance missile, which has an extraordinary 357:1 throttle range with 15% loss at the low end.
I found a poorly labeled and confusing graph that suggests that the CECE upper stage engine suffers about 5% when throttled far down.
If you cared to, you could design an "afterburning" rocket engine, dumping extra fuel or oxidizer (or any other working fluid, for that matter) into the nozzle; this would give you a big thrust boost, cooler (and possibly dirtier) exhaust, and a large loss in efficiency. Thus power could be increased beyond the point of peak efficiency.
(Apparently AJR has patented a variation on this, injecting both fuel and oxidizer into the nozzle -- effectively using the upper part of the nozzle as combustion chamber, for a more appropriate expansion ratio at sea level, apparently?)