Will changing gain affect frequency response?

I don’t believe this has been expressly mentioned. You can significantly effect how you "drive" your speakers by adjusting the gain and therefor the volume the speakers put out. I would be Very cautious however about over-driving the speakers and burning out the crossovers, blowing out the speaker surrounds or similar speaker or other damage. A friend of mine blew out the speaker surrounds of his speakers 3 times before he caught on that he had the gain adjusted too high. Adjusting the gain too high can also cause your equipment to "clip". The best setting for gain is just before your equipment begins to clip. This is the point at which you are driving your speakers and your equipment to the max - which is what you want to do- but you are not over driving it. I understand all of the discussion of volume and our perception that louder is better. Gain is really a different issue although it also results in variations in volume. Again, I would be most cautious about turning the gain up with careful consideration of the overall effect it is having on all of your equipment.

In addition to the points already made, an additional issue is that a room can sound brighter at higher volume levels!

I run into this all the time at audio shows where the rooms are hotel rooms that aren't built as well as rooms you might have in your house. At lower levels reflective surfaces such as windows and framed pictures might seem innocuous but if the room is driven too hard you might find it painful.

In a nutshell the room is at least 50% of the total system quality! I have lots of LPs and they have proven to be good room diffuser elements when neatly stored in their shelves (rather than stacked on the table beside by my turntable...). This is why there is a room treatment business.

One other thing that can affect tonality when the volume is increased is the effect of the volume control on the surrounding circuits. This usually affects tube preamps more than solid state. There is this thing called Miller Effect, which is the input capacitance of the amplifying device interacting with the outside world. It can affect frequency response! A classic example is a loss of high frequencies in the middle of the control (the impedance of the control interacts with the input capacitance of the tube or transistor) If you happen to use that part of the control a lot, turning it up past that point might make for more HF response.

This problem can be controlled by careful design.

I own a Mark Levinson preamp that has adjustments in gain for each input, the values are 0,6,12 and 18 db. My a amp is a McIntos h. I have played with the adjustable gain settings and my system comes to life with the preamp gain for my CD unit at 12db. 0 sounds bright, thin, lacking dynamics, 6 sounds much better but 12 really gives me the dynamics,  bottom end, much better mid-range, that the settings of 0,6 just cannot match, so gain does impact how your amplifier performs and it does big time.