The K1 is particularly hot. Anyone who thinks that any other K is hot would be surprised to find out how much hotter the K1 is for the rider.
The main cause of this is the bike's aerodynamics, the fairing (especially the front wheel fairing) is designed to re-direct the air stream around the outside and not through. In addition the rider has more protection from that air stream than any other K series.
The engine heat builds up inside the fairing and get pushed straight onto the rider's legs when the fan cuts in (which it does much more frequently than any other K I've owned). My mate rode my first K1 just after I'd collected it. It was a fine day and the bike was warm. He actually thought that the radiator or a hose had split when the fan kicked-in.
There are two things that can be done. BMW recognised the issue after the bike was launched, they offered a heat shield arrangement that fits inside the fairing behind the radiator ducts.
Parts 10, 11,12,13, & 14 here:
The parts are still available and not too pricey. Fitting does require a major strip down though.
The second thing is to make certain that the radiator ducting is correctly fitted. The main shroud is rarely the problem, it's the two side ducts that connect to the shroud. These are a push fit and need to be removed for some servicing processes. The result is that they are often not re-fitted properly and when the fan kicks-in the hot air isn't exhausted through the fairing side vents but comes back onto the rider's legs.
You'll know if the ducts are attached properly because even in winter, in towns or when travelling at low speeds, it will still be very hot if they are incorrectly fitted.
The ducts are fiddly to seat properly, I've found that the best way is to slip a hand inside the duct, feel for the edge of the radiator shroud to guide the neck into the main radiator shroud while clicking it home by pushing with the other hand.
With the heat baffles and the ducts fitted properly mine is a lot cooler than my first K1 (which didn't have the baffles), even in this weather. I rode my first K1 around Spain for three weeks in high summer back in the early nineties. It was very hot but I think that I didn't notice it much in the cities because I was so intent on not being taken out by the local drivers!