First of all, one can compare different apps directly only if you have exactly the same configurations for all 3 apps as concerned to sync, push, etc.
(And I am sure they are not the same. E.g. Gmail is not using IMAP, and, at least in my version, there is no option to configure frequency of synchronization. And their push is not IMAP push.) Do you have exactly the same set of labels (in Gmail)/folders (in other two apps) synchronized? Do you have the same settings for how many messages are being synced?
E.g. default in Aquamail is in the number of messages cached/synced (By the way, what are your settings there?), while Gmail sets the number of days instead. Depending on your account usage pattern, those could effectively mean very different number of messages to be synced.
All those parameters make a difference in how much battery is used.
And then, there are some factors related to how different apps are working. But before that, I should make a disclaimer: I am not a developer.
K-@: I don't know much about k-@ (except the description I've read on Google Play's page). I am not sure if they are using cloud storage as some other apps do. And their website is not available. (It says: "Published: Friday, 26 February 2016 16:28": "We switched hosting provider and are in the process of setting up a new website. Stay tuned." -- That's over half a year... I would be very much concerned trusting my e-mail to that app, but that's a separate issue.)
As far as I understand, use of cloud storage can reduce battery and mobile data usage, but it is a privacy/security problem: it likely means that your credentials (login/password) and messages are stored on that company's server. (As an aside, if the company cannot maintain their website, can you trust them with the proper maintenance of the cloud server/storage?)
As for Gmail, - I am not surprised: Gmail is using its own e-mail protocol (developed specifically for the mobile app, and its own OS internal features. For comparison, IMAP is a rather old e-mail protocol. It was developed well before mobile devices. So, despite recent patches/extensions, it is not optimized for mobile communication. (But, it is a rather solid standard shared by many different e-mail providers. And that's its strength.) So, if Gmail didn't use less battery, I'd be surprised.
However, I can tell you that I had some Gmail messages showing up on my phone several hours after arrival. I know a few other users who had a similar complaint.
I hope this information is helpful to you.
Aquamail developer, Kostya, might have more detailed insights once he is back.
