Pearson, whose side are three points adrift at the bottom of the Barclays Premier League table ahead of a trip to the fourth-placed Irons on Saturday, has accepted a Football Association charge of using insulting words towards a fan.
The Foxes boss had refused to apologise to the supporter despite a video emerging of the incident during his side's 3-1 home defeat to Liverpool on December 2.
Allardyce - himself the subject of boos from some of the West Ham faithful last season - believes Pearson should be commended for taking the east Midlands club back up to the top flight following a decade in the wilderness, which also included a spell in League One.
"There is a turning point because what happens at football clubs, expectation grows, people think you are just going to be able to go in there and be successful from a period of great joy," said Allardyce, who took West Ham up via the play-offs in 2012 and successfully retained their Premier League status.
"It is the fact that instead of watching your team win every week, it is a big adjustment for the fans when they find out it is not that easy any more.
"I suppose the expectation was built even higher when they beat Manchester United 5-3 and of course the Leicester fans have not remembered where Nigel has taken the club from and to in a short period of time, so they should show him a bit more patience and a bit more loyalty."
West Ham, who have lost just once in 10 matches, will see their ambitions tested with Christmas fixtures against Chelsea and Arsenal, so Allardyce knows the importance of a positive result on Saturday.
"Leicester are a hard team to break down, they work very hard for each other, despite their run of games without a win, they haven't been turned over by anyone, it's only ever by the odd goal, so there are a lot of dangers in Nigel's teams that we need to be well aware of," he said.
Midfielder Mark Noble is sidelined by an Achilles problem, but could feature next week when Allardyce will make full use of his squad.
"If we come out of this period with a few more points, then we will reassess our aims for the rest of the season," he said.
Allardyce remains "comfortable" with the plans to talk about his contract later in the season.
He said: "I only spoke to [co-chairman] David Sullivan over some other matters in the last couple of days
Nothing to do with me or my contract.
"We will sit down when we're ready and comfortable and discuss the situation, it hasn't changed from the start of the season to now."
Reports have suggested Senegalese striker Diafra Sakho is set to have his four-year contract renegotiated, following an impressive start to life at Upton Park.
"I don't see why we should be discussing any contracts with anyone too much in this period when we are playing so many games in such a short period of time
But yet again players don't see it that way, do they?" Allardyce said.
"If they think it is time for them, they don't give a monkeys about anyone else, they just get on with their own thing.
"They won't worry about the fact that we have got so many games and we have got to organise this and that."
England under-21 midfielder Ravel Morrison has just returned to West Ham after having his loan spell at Sky Bet Championship side Cardiff cut short.
Allaryce said: "Obviously it hasn't worked at Cardiff, so we will have to look at the situation over the new year, but my focus isn't on Ravel, it is on my first-team squad continuing the great run of form they are on at the moment."
Source : PA
Source: PA