“Two and barbecue” (or “two and barbeque”) is a phrase that is probably most associated with baseball’s College World Series. The event is double elimination—lose two games and you’re out. In the college NCAA basketball championships, it’s single elimination (“one and done”). “Barbecue” was added, perhaps, because it rhymes with “two.” Lose two baseball games and a team might as well go barbecue. “Go fishing” or “play golf” are sometimes mentioned when an athletic team’s season ends.
   
The “two and barbecue” expression has been used since at least 1987 and is popular in Texas.
   
   
Wikipedia: College World Series
The College World Series or CWS is a baseball tournament that is the culmination of the NCAA Division I Baseball Championship, which determines the NCAA Division I college baseball champion. It takes place in June of each year.
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Format changes
Through 1987, the College World Series was a pure double-elimination event. The format was changed in 1988, when the tournament was divided into two four-team double-elimination brackets, with the survivors of each bracket playing in a single championship game. The single-game championship was designed for network television, with the final game on CBS on Saturday afternoon.
 
Double-Tongued Dictionary
two and barbecue
in the phrase go two and barbecue, to be defeated in the first two games of a double-elimination tournament.
Citations:
1988 Mike Baldwin Saturday Oklahoman (Oklahoma City) (May 21) “Pacific Blanks Kearney Stout Slams Solo Homer in 9th” p. 23: The Lutes were 0-2 and out the past two years at nationals. “Lose two and barbecue. That’s what we called it.”
1991 Austin American-Statesman (Texas) (Aug. 8) “Sports notes” p. 3: “We went West with great expectations,” said Charles Gurkin, president of the Austin Silver Travlers girls 16-and-under fast-pitch softball team. “But once we got to Midland, Texas, it was take two and barbeque.”
1996 [James Cox] Usenet:  rec.sport.baseball.college (May 27) “Re: Question of Sports Dominance”: That’s not the word I’d choose if my school had lost a bowl game by a score of 41-0, been bounced from the NCAA basketball tournament in the first round, and turned in seven straight “two and barbecue” performances in Omaha.
1998 [Jim Carr] Usenet: rec.sport.baseball.college (May 23) “Re: Final: Usc Vs. Clemson”: The current version of that page, at…/conf-rank-98.html, shows quite clearly who went “oh and two and barbeque” in the first two rounds of the tournament.
2006 Michael Murphy Houston Chronicle (Texas) (May 24) “UT tries to regain stature”: Gone is the double-elimination (“two-and-barbecue”) format, replaced by the pool-play setup first proposed by Garrido, one that ensures at least three games for every team.
2006 TexasSports.com (May 26) “Big 12 Tournament chronicles: To the power goes the glory” (in Oklahoma City, Oklahoma): Going “two and barbecue,” which meant losing two consecutive games in the traditional double-elimination tournament format, was always disappointing, but long-time observers knew that was “just something that happens.” Two straight losses is a slump, three is a losing streak.
     
1 July 1987, Anchorage (AK) Daily News, pg. B1:
Players call that unfortunate circumstance, “Lose two and barbecue.” 
 
Anchorage (AK) Daily News
LONG SUFFERING TEAM DISCOVERS WINNING
Author:  Beth Bragg
Date: July 17, 1988
Publication: Anchorage Daily News (AK)
Page: D3
Word count: 869
Something was wrong. It was the opening weekend of the softball season, the first tournament of the year, and the guys from Humana Hospital weren’t barbecuing. Barbecuing is what almost every respectable softball team does after being eliminated from a tournament. Lose two and barbecue. It’s a softball tradition right up there with fighting for a parking spot on the park strip or finding an empty table at Pierce Street. 
   
29 July 1996, Washington (DC) Post:
A “one-two barbecue” is what happens when you lose two games in a double-elimination tournament, and you may as well go get dinner.
   
Google Books
The Legendary Barons
by Michael J. Vaughn
Dead End Street
2003
Pg. 71:
We suffer a quick loss, and face the dreaded “lose two and barbeque.”
   
ESPN.com
Monday, June 21, 2004
Updated: June 22, 2:35 PM ET
Hold the Tigers
By Jim Caple
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LSU missed the CWS in 2001 and 2002 before returning last June, only to be eliminated in two games—what is known around here as going “two and a barbecue.’‘
     
Atlanta (GA) Journal-Constitution
‘Big Baby’ erupts for LSU
By CARROLL ROGERS
The Atlanta Journal-Constitution
Published on: 03/17/06
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“I think it was the will to win and the determination I had to win,” Davis said afterward. “I kept thinking about the barbecue. I don’t want a barbecue right now. They got tired and realized how big I was. They’ve got to bang all day.”
   
Davis was borrowing the “lose two and barbecue” lingo from the College World Series. LSU, which traditionally has more to celebrate in baseball and football, can boast a little basketball…