I believe this is more folklore than science. A significant color shift happens only for a couple of minutes during sunrise and sunset. The change in brightness is probably significant, but it is hard to believe that the color has a significant physiological effect (but the placebo could be very strong!). In my experience, f.lux and co. make it pretty difficult to read text due to the low contrast, and simply changing the screen brightness is much more effective.
> Further analysis of these 15 reports indicated that a two-hour exposure to blue light (460 nm) in the evening suppresses melatonin, the maximum melatonin-suppressing effect being achieved at the shortest wavelengths (424 nm, violet)
> The melatonin concentration recovered rather rapidly, within 15 min from cessation of the exposure, suggesting a short-term or simultaneous impact of light exposure on the melatonin secretion.
> A significant color shift happens only for a couple of minutes during sunrise and sunset.
This seems like a very dubious claim - the "golden hour" is obvious to everyone, and there's an intuitive mechanism for sunlight being "warmer" in the morning and evening (blue gets scattered in proportion to the amount of air it travels through). Do you have a citation for this?
What's true for outdoor lighting is just as true for indoor lighting:
> It is crucial to control upward-directed light, but we now know that the color of light is also very important. Both LED, and metal halide fixtures contain large amounts of blue light in their spectrum. Because blue light brightens the night sky more than any other color of light, itβs important to minimize the amount emitted. Exposure to blue light at night has also been shown to harm human health[1] and endanger wildlife[2]. IDA recommends[3] using lighting that has a color temperature of no more than 3000 Kelvins.