Nice! I'd say a good counter would be refraining from 4. ...d4
Kings Gambit Powerful Variation

Better is 7...Qxg4 with a clear advantage to Black.
9.Qh5 is a mistake. 9.Nxf7 is a forced draw, and 9.Nc3!? is an interesting way to play for a win.
11.Bc4 seems wrong- white should have played 11.Nc3. After 11...Nf6! Black will keep some material advantage, although converting it is not easy.
12.Kf2?? loses to the simple 12...Be7.
And so it goes...

4.h4! would have kept white out of trouble. Steinitz, Rubinstein, and other positional players were/are of the opinion that 4.h4! favored white and the game demonstrates why 4.d4? is premature here. 4.Bc4 is also premature on account of 4...Bg7! when black has ...h6 depriving a potential h4 thrust of its purpose.
However, if black understands the theory at all he'd either go for a Falkbeer Countergambit (1.e4,e5 2.f4?!,d5! 3.exd5,e4! exerting a cramping influence upon white and rendering the f4 pawn useless) or a Fischer Defense, details of which can be found here:
http://thenicl.wikispaces.com/file/view/A+Bust+to+the+King%27s+Gambit+-+Fischer+-algerbraic.pdf
And the Falkbeer being a strategic refutation of the King's Gambit can be found here:
http://reluctant-messenger.com/chess/ebooks/Masters%20of%20the%20Chessboard.pdf
Mentions in Anderssen's bio near the beginning that the Falkbeer strategically refutes the King's Gambit.

4.h4! would have kept white out of trouble. Steinitz, Rubinstein, and other positional players were/are of the opinion that 4.h4! favored white and the game demonstrates why 4.d4? is premature here. 4.Bc4 is also premature on account of 4...Bg7! when black has ...h6 depriving a potential h4 thrust of its purpose.
However, if black understands the theory at all he'd either go for a Falkbeer Countergambit (1.e4,e5 2.f4?!,d5! 3.exd5,e4! exerting a cramping influence upon white and rendering the f4 pawn useless) or a Fischer Defense, details of which can be found here:
http://thenicl.wikispaces.com/file/view/A+Bust+to+the+King%27s+Gambit+-+Fischer+-algerbraic.pdf
And the Falkbeer being a strategic refutation of the King's Gambit can be found here:
http://reluctant-messenger.com/chess/ebooks/Masters%20of%20the%20Chessboard.pdf
Mentions in Anderssen's bio near the beginning that the Falkbeer strategically refutes the King's Gambit.
Aw, really?
4.h4 is the mainline of course, when white with accurate play can equalize. John Shaw in his recent KG book advocates 4.Nc3, which IMO is not as good, but certainly interesting.
2...d5 3.exd5 e4? is a mistake, which gives white the advantage after 4.d3. The right move is certainly 3...exf4.
The "Fischer defence" is something that Fischer never played OTB- it's rather his wannabe KG refutation, published in an ancient article. It is almost certainly NOT the best way to deal with the King's Gambit.

Reti in his Masters of the Chessboard gave clear and concise logic as to why 3...e4 is good though (I'm not saying one side is right or wrong but 3...e4 seems quite reasonable especially in light of Reti's reasons).
3...exf4 would give white the open f-file he wants. The idea behind Fischer's Defense is allowing white an extra (useless) tempo to take the potential for Ne5 away (which is quite strong for white in the Kieseritzky according to Reti).
The King's Gambit seems to lead to tough endgames (especially with rooks involved and the kingside structure is too committed) for white where every tempo counts. One can't speak too generally about the King's Gambit however since there are so many different variations with their own flavor, Falkbeer, Cunningham, modern, KGD, Fischer's Defense, and of course ...g5 will carry with them different middle game and end game strategies.
The cramping black f4 pawn in any event is a complication that white can do without, the Fischer Defense has black's strengths in the ...g5 variation with none of the weaknesses on account of Ne5 being unplayable.

10...Be7. I can't see any mate or perpetual and White is down material.
How does that refute 11 Qxf7+ Kf8 12 Bg5! if Bxg5 its still Qf8=MATE
If 11 Nf6 Bxf6 12Bxf6 Qxf6+ forking the rook.

10...Be7. I can't see any mate or perpetual and White is down material.
How does that refute 11 Qxf7+ Kf8 12 Bg5! if Bxg5 its still Qf8=MATE
If 11 Nf6 Bxf6 12Bxf6 Qxf6+ forking the rook.
12.Bg5 Qh1+ draws (the king cannot go to d2 because the g5 bishop is grabbed with check), but 12...c6 seems stronger- white will take back the rook, but his position does not inspire much confidence (drawable at best).
9.Qh5 is (almost) a lemon.

12.Bg5 Qh1+ draws (the king cannot go to d2 because the g5 bishop is grabbed with check), but 12...c6 seems stronger- white will take back the rook, but his position does not inspire much confidence (drawable at best).
9.Qh5 is (almost) a lemon.
I missed the check, you are right, now I have to come up with something new.
I played this as white and its my fav Kings Gambit attack, outright destructive. I don't know what its called though but here it is. I use it all the time as I have not seen other players effectively counter it yet.
Kings Gambit Accepted - Unknown Variation
1.e4 e5
2.f4 exf4
3.Nf3 g5
4.d4 g4
5.Ne5 Qh4+
6.g3 fxg3
7.Qxg4 g2+
8.Qxh4 gxh1=Q
9.Qh5 Qxe4+
10.Be2 Qd5
11.Bc4 Qe4+
12.Kf2 Qxd4+
13.Be3 Qd6
14.Qxf7+ Kd8
15.Bg5+ Be7
16 Qf8++