This can be done quite efficiently with tapply. I've altered your data somewhat by duplicating teams' games, with random scores and dates. This takes the mean of the most recent 2 games, as specified in the tail function.
# create some data
d <- structure(list(Div = structure(rep(1L, 33), .Label = " E0", 
  class = "factor"), date = structure(c(15013, 14990, 14996, 15001, 14995, 15006, 
  15020, 15032, 15023, 15022, 15015, 15016, 15034, 14994, 14986, 14998, 14982, 
  14979, 14980, 15016, 15031, 15013, 15031, 14999, 15025, 14978, 15007, 15026, 
  14992, 14997, 15023, 14986, 15028), class = "Date"), 
  value = structure(c(3L, 4L, 5L, 7L, 8L, 11L, 9L, 10L, 6L, 1L, 2L, 3L, 4L, 5L, 
  7L, 8L, 11L, 9L, 10L, 6L, 1L, 2L, 3L, 4L, 5L, 7L, 8L, 11L, 9L, 10L, 6L, 1L, 
  2L), .Label = c("Arsenal", "Aston Villa", "Blackburn", "Fulham", "Liverpool",
  "Man City", "Newcastle", "QPR", "Stoke", "West Brom", "Wigan"), 
  class = "factor"), pts = c(0.5, 0.5, 0.5, 1, 1, 1, 1, 0, 1, 0.5, 0, 1, 1, 1, 1, 
  0.5, 0.5, 0, 0.5, 0.5, 0, 0, 0, 1, 0, 0, 0.5, 0, 1, 0, 0.5, 0.5, 0.5)), 
  .Names = c("Div", "date", "value", "pts"), row.names = c(NA, 33L), 
  class = "data.frame")
# sort rows by date
d2 <- d[order(d$date),]
# mean of all games
tapply(d2$pts, d2$value, mean)
# mean of last 2 games
tapply(d2$pts, d2$value, function(x) mean(tail(x, 2)))
# To tidy up the output, you could use simplify=FALSE and do.call(rbind, x):
# e.g., mean of last 2 games:
do.call(rbind, tapply(d2$pts, d2$value, function(x) mean(tail(x, 2)), 
  simplify=F))
            [,1]
Arsenal     0.25
Aston Villa 0.25
Blackburn   0.50
Fulham      1.00
Liverpool   0.25
Man City    0.75
Newcastle   1.00
QPR         0.50
Stoke       1.00
West Brom   0.00
Wigan       0.50