I don't know anything about made-for-cycling caps, and certainly not at Rapha prices. Except for an Altura Night Vision Jacket I have nothing currently in use that is specifically made for cycling, because I find cyling gear to be particularly badly fitted and overpriced too.
However. I wear a helmet, and I have found, by trial and error:
1. I have a lycra sports running cap that is exceedingly useful. This has no brim and is like a beanie, deep enough to cover the ears. I wear it in the summer under the helmet to keep the sweat out of my eyes, and in the spring and autumn simply to keep my head warm. Close fitting, but not warm enough for the coldest weather. Black, with a non-denominational graphic in pale gray, no maker's name. Bought from Lidl; I'll buy several more if they bring it back. Probably also in sports shops under one brand or another. Love it and wear it more often than anything except my cycling jacket and my fave leather gloves.
2. My helmet has a visor that stays on year-round, but in high summer a bigger bill is often an advantage riding the ridges of the hills that define the valleys of my hilly environment. Then I wear a baseball cap under my helmet. The baseball cap, cotton, is cooler than the lycra item above, but it is rarely warm enough here for that to be a compelling reason to switch to the cotton cap. It has an open back with an adjustment strap; that doesn't bother me, but it would do more to keep the sun off the back of your head if it were closed and had elastic sewn in for self-adjustment. Also bought from Lidl.
3. For the worst cold of the winter, I have several Thinsulate beanies which I wear under my helmet. The one on my head right now (not taken off after taking the first cycle of a rather cold spring day) claims to weigh only 40gr. These beanies range from expensive sports items to cheap supermarket jobs; I see no quality difference except that one of the expensive sports department ones had an unraveling seam from new. The reason I have several is that when it is really cold and windy I cycle in a padded leather jacket, and my rides are exactly long enough to ensure that I don't start perspiring until 200 meters from home, which just happens to be at the top of a steep hill. The beanies need washing more often than anything else I put on my head, so I need more of them.
BTW, my head is the only place I wear lycra; you lose between a twelfth and an eighth of your body heat through your head, and in coolish climes that extra heat retained can double your comfortable ride; I'm not claiming this is scientific, but my experience is that if I forget the hat a shorter loop is advisable but if I'm wearing the hat, the longer loop is possible before I start wondering about frostbitten ears and my energy-level bottoming out. The lycra does well in regulating the temperature on my head in all except extreme cold. Also, as supporting evidence, often winter rides can be relatively longer simply because I (and my pedal pals) are willing to regulate our speed to our respiration rate, and we're thus warm but not uncomfortably so, and therefore the length of the ride is regulated by interest rather than comfort.
