The Hexenhammer's Withering Gaze ability states:
Hexenhammers study the effects of a witch’s evil eye, and can use this dark magic to steel their gaze in a similar manner. Beginning at 1st level, whenever a hexenhammer successfully demoralizes a creature as a standard action, instead of applying the shaken condition, she can apply the effects of the witch evil eye hex as if she were a 1stlevel witch (save DC = 10 + 1/2 the hexenhammer’s inquisitor level + her Wisdom modifier).
The important bit here is that this only works if you demoralize as a standard action, so it won't work with things like Cornugon Smash which demoralize as a free action. My question is about the Inquisitor spell Blistering Invective, which you cast as a standard action and states
You unleash an insulting tirade so vicious and spiteful that enemies who hear it are physically scorched by your fury. When you cast this spell, make an Intimidate check to demoralize each enemy within 30 feet of you. [...]
Would the demoralize from this spell work with the Withering Gaze ability, since the spell is cast as a standard action?
I'd say no. At the risk of seeming an absolute pedant, making an intimidate check to demoralize =/= a standard action to demoralize.
The former is a result triggered by another action; the latter is a specific action you can choose to take.
For comparision, see rules situations about making an attack as a standard action (with some additional effect) vs using Vital Strike, or Cleave.
Blistering Invective is a standard action, yes?
It demoralizes your opponents, yes?
I believe the answer is yes.
But I could be wrong, I just always err on the side of letting people play. Given that trading the shaken condition for a level 1 evil eye hex is not going to break anything, ever.