He remembers the 2004 World Series champion Red Sox, Manny Ramirez once cranking a homer over the Coca-Cola bottle in left field and seeing his favorite players: Nomar Garciaparra, Pedro Martinez and Kevin Millar, among others.
But just like Cobb's favorite players in the past, he's in search of a trip to the postseason and a ring. He'll take the mound for the Rays at Fenway on Wednesday against the Red Sox in the finale of a two-game set.
"There's so many memories just watching the games with my dad and other family members here," Cobb said. "It'll always be a special outing for me whether [the Red Sox] are in the race or not. It's a huge game for us, so that playoff atmosphere will be there for us."
Tampa Bay was on the winning end of Tuesday's matchup, a 5-2 triumph that was its sixth straight victory.
Cobb himself has been pitching well lately, allowing three or fewer runs while lasting at least six innings in three of his previous five starts.
He last pitched Sept. 17, a loss to Boston that saw him give up three runs over six innings on two hits and three walks, but he's taking the mound on eight days' rest.
"It's not easy," Cobb said of the strategy to keep his young arm from piling on too many innings. "The theory of it's a good thought, giving me rest, but it also does complicate things a little bit. But I think I'll be fine."
The Red Sox will counter with Jon Lester, who is coming off back-to-back seven inning starts in which he was saddled with losses.
In the more recent of the two, a loss to Baltimore, the Orioles tagged him for four runs on eight hits and three walks.
Lester has made only one start against the Rays this year, a May 25 outing that saw Tampa Bay rock him for seven runs on six hits and three walks in four innings.
Rays: Maddon shuffles rotation
Tampa Bay manager Joe Maddon decided to flip starters Jeremy Hellickson and Matt Moore in the rotation, meaning Hellickson will take the mound Friday against the White Sox on normal rest while Moore will pitch in Saturday's matchup.
The switch would figure to line Hellickson up to pitch the season finale on Oct. 3 against Baltimore, and Moore's start Saturday will likely be his last of the season unless the Rays make the playoffs.
Hellickson has pitched well as of late, allowing just three runs in his last 16 1/3 innings. Moore, meanwhile, hasn't been as fortunate, dropping four consecutive decisions and allowing nine runs in 9 2/3 innings over his last three starts.
Maddon said the decision is not a reflection of possible pitch tipping from Moore and maintained that he continues to be healthy.
"He can actually stand on the mound and say, 'Here comes my fastball,' and I still think he can be successful," Maddon said. "I mean, all this stuff is way overblown. I want him to stop thinking about that. If he makes good pitches, regardless if they know it's coming or not, I still think he can be successful."
Red Sox: Ellsbury may play Wednesday
Dealing with an injury that has kept him out five straight games, center fielder Jacoby Ellsbury could be back in Boston's lineup on Wednesday, said manager Bobby Valentine.
"He felt good enough to play, and then the training room thought another day would be best, so he's doing fine," Valentine said of Ellsbury.
Ellsbury, who was batting .277 with 26 RBIs in 70 games, requested that Valentine not disclose the injury.
In his usual spot in center, Che-Hsuan Lin got the start, going 0-for-3 at the plate.
Rays catcher Jose Molina is not expected to make the start behind the plate Wednesday.
Red Sox reliever Daniel Bard pitched a simulated inning Tuesday, and Valentine said he expects Bard to pitch at some point before the end of the season.
Boston currently leads the season series, 9-8, between the two teams.