Sort of. The flow rate is the essentially the same regardless of temperature. The first stages work off of pressure differentials, so while you are moving more volume per breath, you can't utilize that in your body, so you're breathing more dense air, but it doesn't really matter because you can't use it. Same thing happens when you go deeper, you are breathing the same volume of air to fill your lungs, but because of the increased density of the water, you are using more air out of the tank for each breath.
If the tank changes temperature your pressure will change per Boyle's law.
I.e. Tank is filled to 3500psi at 75F, but you're diving in 50F water, the pressure will drop to 3350 once the tank equalizes temperature which doesn't take particularly long. This is why when you are diving in cold water it is important to mark your pressures once the tanks have hit the water.
Conversely another common one is the tanks will be at 50f when the shop fills them in cave country in the winter to 3600, and once they hit the nice warm spring water, they will rise to 3800. Much less of a concern in recreational diving, but in technical diving it's a big concern.
The "rule of thumb" is 5psi per *F, but it's VERY loose and only good for large temp swings, 20F+. It's quite a bit more when the tanks are at lower delta T's, but those generally don't matter much, and also is only good for low pressure tanks.
I.e.
LP steel filled to working pressure at 50F has about a 6psi/*F delta when it hits 70F.
A HP steel filled to working pressure under the same conditions, has about an 8psi delta
Formula for that is
(460+Temp final)*(PSI init+14.7)/(460+Temp init)=PSI Final
460 is because you have to work in the Rankine scale to get it to work which has 0 as absolute zero, but the degrees have the same scale as Fahrenheit.
So to answer your question, yes but the opposite of what you said. Because your SAC is independent of gas density, and your tank volume is really only dependent on the tank pressure, if the tank gets warmer you get more bottom time, but if it gets colder, because the pressure drops, you have less cubic footage inside of the tank to breathe, so you have less time. The only temperature that matters is the water temperature you'll be diving at, so ideally you want the tanks to always be colder than the water when you check them for pressure. That means making sure the tank is room temperature before you check the pressure and ask for a top off if they short filled you. In NFL most of the shops will fill to 3700psi or so to let the tanks cool to 3600. Allows the tanks to hit about 90F during filling and cool to 75F room temperature.
