"will biodegradable toilet paper still break down in arid climates such as Southern Utah or Colorado?"
If the cathole is in moist soil, yes it will break down. If the soil is bone dry, then no, it won't. As the previous poster noted, the cathole shouldn't be very deep due to the aerobic microbes' requirements for oxygen. You will need to balance these two issues. One caveat: if the soil is salty (I know some desert soils can be salty) that may impede degradation.
Microbes need 4 things before they can decompose TP: Proper temperatures (hopefully above freezing), a nitrogen source (your excrement has plenty of "N"), a carbon source (TP and faeces are carbon rich but urine is not), and moisture.
I have often thought that burying extra carbon (ie: TP) with a nice "steamy one" could be seen as benefitial from soil and water quality prospectives. The extra carbon provided by the TP helps to tie-up "N" that could be leached down into a shallow aquifer or into nearby surface water (a bad thing). Later, the N (and other nutrients) will be slowly released and hopefully utilized by surrounding vegetation. In addition, the TP coould increase the water available to microbes for degradation as well as mediate fluctuations.
This corresponds with how gardeners and farmers compost their manure. Raw manure is rarely applied as it provides too much N at once.
Anyone care to do some field testing?