I don't know where you saw that warmer water will lose more FC due to sunlight than colder water. That simply is not true. Please link to such posts. Chlorine loss from the UV in sunlight does not depend on temperature because it only depends on the number of photons per area entering into the pool and on the concentration of chlorine. The photons of light are traveling
much faster (at the speed of light) than the molecules containing chlorine so the temperature which relates to the speed of those chlorine molecules is irrelevant. From the point of view of the photons, the molecules of chlorine are essentially standing still so the cross-section of those molecules which is the area with which the photon has a quantum probability of reacting with the molecule is independent of the temperature and only related to the concentration of such molecules in the water.
Now as noted in
this paper there can be a temperature dependence on the subsequent chemical reactions that occur
after the photochemical reaction occurs, but for the breakdown of chlorine by UV, it's pretty much all over with such breakdown since the probability of having the OH• and Cl• reform compared to other reactions that lead forming chloride and oxygen gas is low and not temperature dependent. At most, the intermediate concentrations of some intermediate species such as hydrogen peroxide may be higher at lower temperatures, but by that time it's too late and the chlorine is already on it's way to becoming chloride. Also as noted in
this paper, there can be a small dependence on temperature in terms of the rate of relaxation from excited states so there is a small but negligible temperature dependence on having the HO-Cl vibrational state be excited and not break apart as often because of low temperature, but it takes a very low temperature before that effect would be seen in practice. As noted in
this paper:
When the concentration of free chlorine is low (3.5 mg Cl/L) to moderate (70 mg Cl/L), the quantum yields of HOCl and OCl- are 1.0 ± 0.1 and 0.9 ± 0.1, respectively.
The quantum yield of 1.0 means that every photon that reacts with HOCl results in its breakup into OH• and Cl• radicals while the quantum yield of 0.9 means that 90% of photons that react with OCl
- result in its breakup into O
-• and Cl• radicals. You can read more about photochemistry in
this link.
Chlorine consumption that is related to temperature is for chemical reactions with chlorine such as oxidizing pool covers, bather waste, pollen, leaves, algae, etc. And yes, algae grows faster in warmer water but such consumption won't matter if there is sufficient FC/CYA since algae will get killed before it can reproduce so the rate will be based solely on the rate of blown-in algae spores which is usually not measurable (pollen, on the other hand, can be voluminous as can pods and other material dropped from trees).
A pool with a solar cover that is opaque to UV would lower the loss of chlorine from sunlight that is
not temperature dependent but would increase the loss of chlorine from oxidizing the cover which
is temperature dependent.