You can, and there are many who do! However, you'll find if you play each note separately over each chord, that some will fit better than others. You'll also find that some individual notes will fit to two, and at a pinch, all three chords.
The root note, A, fits well over A7 and also D7, but not so well over E7. The next note, C, fits very well over D7, and in a bluesy manner over A7, although a lot of better players give it a tweak - a bend up, hinting , or even arriving, at C#, the maj. 3rd of A, where it fits perfectly.
I won't take each note in turn - that's your job! You may feel some fit better, or not at all... Obviously, the better fitting ones will be reflected in the prevalent chord, so will work best - on beats 1 and 3, with passing notes which maybe don't belong at all in between.
The other big issue is that, as said, you can widdle away with Am pent. notes all through, but how about Dm pent. notes on D7, etc? You'll actually find some notes from one are also from the other. That's why it works by itself - to a degree.
When you get a bit fed up with just those 5 notes (from A, or from a, D and E), there's a bitter-sweet note that sounds horrible and gorgeous together, it lives between the third and fourth note you play in min. pent. The blue note!!
And, well done for recognising 'guidelines' instead of 'rules'!