As I found out the hard way on a road trip, where temperatures were in the mid 90s, the AC compressors in the third-generation CR-Vs have a very common failure. The compressor itself doesn’t fail, but there is an electric clutch actuator coil that burns out, which prevents the clutch from engaging. One other symptom of this is that the relay is damaged–if the coil fails and draws too much current, it can burn out the contacts in the relay. Note that if your relay fails and you replace it, and it starts working again, there was a reason your relay failed. Keep a close eye on it.
While there is a way to replace only the clutch coil, I feel it’s better to replace the whole compressor. If you replace the coil, there is also the possibility that the clutch plates themselves may wear out, or the compressor itself may malfunction. If you’re doing the work yourself or have a garage do it, you’re repeating much of the same work. (Plus, there is a snap ring deep inside the compressor that is very difficult to get at if you don’t remove the compressor from the car when you replace the coil–many DIYers have reported that this step was the most frustrating in the whole process.)
As I had the dealer do both of our CR-Vs, I had them both use remanufactured Honda compressors, as they came with a 3 year/36,000 mile warranty nationwide at any Honda dealer in the US. Doing it myself, I probably would use a Denso remanufactured compressor.