Only with Improved Awesome Blow
There's a lot of confusing language in the manual for this one. Most importantly, these three concepts are something different:
- Attack (action): This is an action you can take, normally as a standard action. If you have a BAB of 6 or higher, you may make multiple attacks as a Full Attack. There are 2 Combat Maneuvers (Sunder and Trip), which say "in place of a melee attack". These can be made as part of an attack action, but they are not attacks, as they are in place of an attack.
- Attack Roll: An attack roll is part of the attack action, or of other actions that specify they use it. Some things may require an attack roll without being an attack action. Examples include spells like Scorching Ray(which requires a Ranged Touch Attack), or actions like combat maneuvers.
You must declare that you are using this feat before you make your attack roll (thus, a failed attack roll ruins the attempt). Stunning Fist forces a foe damaged by your unarmed attack to make a Fortitude saving throw (DC 10 + 1/2 your character level + your Wis modifier), in addition to dealing damage normally.
Stunning First refers to both an attack roll, and your "unarmed attack". Combat maneuvers aren't attacks unless they say so. Improved Awesome Blow says "the brawler can use her awesome blow ability as an attack rather than as a standard action." This makes it an attack, and therefore Stunning Fist can be used in conjunction. Normal Awesome Blow never mentions anything, and therefore, does not allow Stunning Fist.
In similar vein, Pummeling Style and Combat Rhythm would function with Improved Awesome Blow only.
Similarly, Shield of Dawn would damage you, but that could be avoided by spending a Standard Action instead of an attack. ("...the brawler can use...", so no obligation to do so.)
Rules as Written vs Rules as Interpreted
Personally, I consider this an oversight in the rules. It think they intended to have Improved Awesome Blow replace the attack action. If interpreted that way, none of these feats would function with Improved Awesome Blow.
That would bring it in line with Trip and Sunder, which use "in place of a melee attack", instead of "as an attack".