I have been lately experiencing similar issue with one difference. In the morning, the cold engine starts right up.
When restarting the hot engine, it will crank for a while and once it finally fires up, it initially idles fairly rough.
No CEL, haven't checked for codes. No gas smell.
At first I thought the starter was getting stuck, but that does not seem to be the issue.
Because if I turn the key to position-II and wait for about 3 seconds until all lights go off and beeping stops, then it fires up no problem.
It reminds me of one of my motorcycles which requires to wait with the key in position-II until the fuel pump gets primed, and you can actually hear the pump making noise.
With my Pilot there is no problem when the engine is cold, so gradual loss of fuel pressure over time seems unlikely.
Vapor lock crossed my mind, but that doesn't seem to be an issue with Pilots and injected engines in general.
Overall very similar to what the OP is experiencing, except in my case it happens only when the engine is hot.
So far I have installed a new battery, cleaned the MAF sensor and the EGR valve, without any change.
Low mileage 2011 Pilot - 54k. VCM muzzled at 38k. Stock spark plugs.
Only use top tier 87 and Mobil-1. Mostly city driving, lately about 4k/year.
The next steps will be along with what Nail Grease suggested: adding Techron, italian tune-up and a new PCV valve.
How to diagnose a leaking injector?
Thanks