During single engine operation the APU will definitely be started, if only to provide power/bleed air in case the remaining engine also stops. I believe all ET

During single engine operation the APU will definitely be started, if only to provide power/bleed air in case the remaining engine also stops. I believe all ETOPS planes need to be able to start the APU at maximum certified altitude. Interestingly, until recently the 737 (which was not designed with ETOPS in mind) had to leave the APU running for the whole duration of ETOPS flights. It took a few years of data gathering before the FAA was sufficiently convinced that a cold-soaked 737 APU would start at cruise altitude. They now are, and "APU On-Demand" is now used on 737 ETOPS flights (i.e. Alaska Airlines flying west coast US to Hawaii)