I've figured out that the switching buck converter would have been what I needed - thank's for the tip on that! But I do not have such converter and I will not get my Christmas light done in time if I should order one for the project. Now I have taken a power adapter from my RbPi, it is rated at 2A @ 5.1V, and I have measured it to 5.3V. This is in the green area but I have put in the before mentioned regulator which gives me a nice 4.25V which is will ensure my LED's not burning out too fast - this works very well! The regulator only has to dissipate a maximum of 1.6 watt and it does only get slightly warm and nowhere near as hot as before. Nice! 
@TomGeorge: The LED's are SK6812 SMD5050 rated at 3.5-5.5V (aka. NeoPixels). They did not burn out at all, they are pretty rugged as long as they do not get too much heat during soldering! I have placed a 1000uF capacitor on the regulator's output. The LED's are soldered into 2 strips (2x10), each LED has a 50Ohm resistor on VDD and the data signals each have a 330Ohm resistor attached as recommended. They are controlled using the AdaFruit NeoPixel library and other than that there's not much other to it
Thanks for you help, guys! I learned a usefull lesson about linear voltage regulators and how (not) to use them