That's why it's called a 'caution'. That said, some competition levels have played around with the idea of an 'orange' card with a sin bin, or just the sin bin, to remove players from a match temporarily and have that team play down a man. Similar to hockey. I've seen sin bins go from 5 minutes to 10, though I think the latter is a bit heavy-handed.
I know that the IFAB have played around with this idea a few times, but to be honest I don't see it being implemented at that level any time soon. edit: should clarify that IFAB is the group that maintains the laws of the game, not FIFA. FIFA is part of IFAB and has 50% of the voting interest, I believe.
A caution, by its very definition, indicates that a player has committed an offense and should not continue in the same way. I would be fine with the sense that having issued or not issued a caution does not change the interpretation of the rules and the leeway afforded by the referee for infractions.
But that's not how it goes in practice -- typically, having shown a card to a player, the official is hesitant to show another card. Thus, by having done one bad thing, a player gets to do more and/or worse bad things than they otherwise would! This makes no sense! If the standard must change, it should go the other way -- a caution would, again as the word plainly means, indicate that the player has already been warned and is now on thin ice, such that any further misdeeds, even relatively more minor, will result in an escalation of enforcement.
People will argue that the harsh penalty of ejection of the player and forfeiture of their roster position on the field justifies making it harder to earn a second yellow versus a first, but I still think that runs counter to the logical consequence that being booked should have. But, because of this reality, I do support some "orange" card, whether it be reducing the team by a man for a temporary duration or ejecting the player but retaining the ability to field a full 11 -- so long as this penalty would rid us of the awful "you were bad, so here's a license to be worse" status quo.
After an accumulation of fouls, the referee warns the captain that the next one will result in a player being sent off for 10 minutes. And they are not reluctant to follow through. The next ruck infringement, even if it is minor and by a new player that has not committed fouls before…off he goes for 10 minutes. And everyone accepts this as the proper way to officiate the match.
And the weaker-willed referees at the lower levels. But I make it clear to players I've cautioned that they are on very thin ice, unless the caution was for dissent. In which case, I just expect them to shut up.
But that's a IFAB/FIFA guidance issue along implementation, not the laws themselves. Law 12 doesn't account for what you're referring to.
That's the point of a caution for persistent infringement, right? You break the rules multiple times, you get a caution. At that point you are already persistent, so additional transgressions would be even more persistent.
But there's no 'additionally persistent', and, generally, you'll never see a second yellow for PI because the player had already checked their behavior because of the original caution.