So, this is just a clarification about how a character's AC interacts with the Prone condition aside from the Flat-Footed condition. Mostly because being prone gives you the ability to Take Cover and some people I'm playing with understand that being prone gives you standard cover against ranged attacks even without taking cover. I'm not sure about that, because the way I read it being prone gives you the ability to gain greater cover - with the mentioned action - even without an actual cover but it does NOT give you normal cover automatically. Can anyone confirm or disprove?
Prone does not grant standard cover
The prone condition states that:
You can Take Cover while prone to hunker down and gain greater cover against ranged attacks, even if you don't have an object to get behind, gaining a +4 circumstance bonus to AC against ranged attacks (but you remain flat-footed).
As you've already noticed, the prone condition doesn't state that it grants standard cover, so it just doesn't.
Normally you need standard cover to get greater cover, but prone lets you do something that you wouldn't otherwise be capable of, which is a rather common thing for rules to do.
And that's really the crucial point, but there's other evidence that points to the same conclusion.
The notion that you don't need "an object to get behind" is strong indication that having standard cover is neither required to take cover in this way, nor implicitly given.
If you are prone, you don't have to take cover as described in the prone condition, you could take cover normally instead:
Requirements You are benefiting from cover, are near a feature that allows you to take cover, or are prone.
You press yourself against a wall or duck behind an obstacle to take better advantage of cover. If you would have standard cover, you instead gain greater cover, which provides a +4 circumstance bonus to AC; to Reflex saves against area effects; and to Stealth checks to Hide, Sneak, or otherwise avoid detection. Otherwise, you gain the benefits of standard cover (a +2 circumstance bonus instead). This lasts until you move from your current space, use an attack action, become unconscious, or end this effect as a free action.
If prone were to automatically grant standard cover, there would be no reason to ever take cover as described in the prone condition, because you could just take cover normally instead to gain a +4 bonus to AC (not just ranged), reflex, and stealth, which is strictly better.
When there are no obstacles (such as ranged attacks made pointblank or from a vantage point), it doesn't make much sense for a prone creature to automatically have cover.
Whereas, when there are actual obstacles, automatically granting cover to a prone creature is unnecessary, because the GM should be granting cover anyway according to the rules for cover. Moreover, the GMG urges the GM to consider cover 3-dimensionally, so the GM can certainly grant some/more cover a prone creature, depending on the terrain and position of foes.
The ruling that the Prone condition provides cover may stem from the pre-requisite for the Take Cover action:
Requirements You are benefiting from cover, are near a feature that allows you to take cover, or are prone. (Nethys - Take Cover, Emphasis mine)
The rules for Cover also state:
When you're behind an obstacle that could block weapons, guard you against explosions, and make you harder to detect...
[If you are benefiting from Cover...] You can increase this to greater cover using the Take Cover basic action, increasing the circumstance bonus to +4.
Cover is normally granted by your surroundings and does not require the Take Cover action. Things like walls and other creatures automatically grant you the benefit of Cover. This allows you to use the Take Cover action in order to upgrade into Greater Cover. It is a reasonable reading, then, that being Prone allows you to meet the condition of the Take Cover action (by providing Standard Cover). This case, though, is likely one of Specific beats General: The specific rules of being Prone explicitly allow you to use the Take Cover action. This is also indicated by the "...or Prone" within the prerequisite.
Usually, the GM can quickly decide whether your target has cover. (Nethys - Cover)
Worth noting, because being Prone also inflicts Flat-Footed, Cover would provide +0 to AC and the Greater Cover you can achieve RAW actually grants a net +2.
You take a –2 circumstance penalty to AC. (Nethys - Flat-Footed)
Personally, though I believe the RAW answer leans towards not, when I've run P2e I do grant Standard Cover against ranged attacks when Prone, but ultimately this means no bonus (or penalty) anyway. This ruling may have a negative interaction with players that take Ranged Trip.