ALL engines 'like' to be worked hard!
Engines that don't get 'worked' generally develop glazed bores and develop high oil consumption and/or contamination from blow-by (spoken from 30-years experience in petrol/diesel engine (50 to 2000 BHP) design, development and test).
Now, one of the issues I have with the RS3 is that when I'm driving in 30/40/50mph restricted zones with the gearbox in 'DRIVE', the gearbox drops into 6th and/or 7th gear ratios far too easily and the load on the engine is greatly reduced - see paragraph one.
So I generally use the 'SPORT' mode in such restricted zones as this holds the gearbox in the 3rd and/or 4th gear ratios I'd normally expect with a manual gearbox. The added benefit for me is the car 'feels' more throttle/speed-controllable and the engine is working just that little bit harder.
OK, fuel consumption will be a little higher (mine has done 25.6mpg from new) but if the engine could talk, it would be a lot happier.
Note: I do this both with my 7-speed/DSG RS3 and 6-speed/DSG TT: the engine oil stays clean with consumption ZERO between changes. Neither do I have coolant loss issues.
My advice: work those RS3 engines a little harder.