RO can be subbed with distilled water, or any water that has very low dissolved solids (under 10 PPM) and does not have added content such as chlorine or chloramine. Other instructions about water temps (and others) can be found here;
Actually, this may be bad advice.
The principal reason that bud washing produces better results is it removes bacteria and fungi from the surface of the plants. With fewer bacteria and fungi, less rotting occurs as it dries and during storage.
It is important to use tap water with chlorine in it to kill most of the fungi and bacteria that wash off. If you use RO, or boil the chlorine out of your water, then you are creating a perfect bath to evenly spread bacteria and fungi to each and every bud. You would need to add something like peroxide to your wash water if you don't want to use chlorine. Wash tubs need to be filled with a solution toxic to bacteria and fungi for this process to work properly. Tap (potable) water kills most bacteria while, generally, being safe for us to consume.
Doc's method of washing is completely consistent with most commercial processing of our dried herbs and spices we eat. This process is not controversial in the herb and spice industry. The FDA provides lots of good info on this process and why it works.
The FDA recommends that harvested plants should be first washed in tap water to remove large foreign materials like soil, insects, their feces, dust (skin cells), etc. Then, a soak in some sort of aseptic bath (like water with lemon juice) should be used to kill surface bacteria and fungi. One or more final rinses in tap water are used to remove traces of the aseptic solution. They also say cold rinses are more effective than warm bath rinses at removing bacteria. Drying should occur in a dark room with still air to minimize airborne contamination.
The FDA says it's important to clean and sterilize your washing equipment between harvests. All tubs should be scrubbed with detergent and a strong chlorine rinse and allowed to dry. Never reuse old wash water for washing.