The England captain has managed just 145 runs at an average of 24.16 in the Ashes series so far, even though his side have already retained the urn after establishing a 2-0 lead with two games to play.
Hoggard, a member of the England side which beat Australia in the memorable 2005 series, recalls times when Cook has struggled previously and is confident the Essex man will soon return to his best.
He told talkSPORT: "Cook went through a patch where he didn't know where his off-stump was and he kept on nicking off.
"Everybody was questioning his technique and asking if he should be at the top of the order but Alastair Cook's strength is his mental strength and he's been working really hard with Graham Gooch, who he knows from Essex, and he's such a solid character he will find a way of getting through this.
"Whether he's having a fantastic innings where everything's easy on the eye or struggling for runs he has that mental toughness and a will to win."
Rain at Old Trafford on Monday led to the the third Test being drawn although Australia will have been encouraged by an improved showing.
And Hoggard is expecting more of the same from the tourists' pace attack when the fourth Test starts at Chester-le-Street on Friday.
He added: "We always knew Australia were capable of taking 20 wickets and their bowling line-up has held its own all the way through the series which shows how good their seamers are.
"The times I've played in Durham there is always a bit of seam movement which will bring back the Aussie bowlers again.
"The England batsmen haven't been the best and the top order has struggled at times, being 40 for three or 30 for three in too many innings.
"I can't just say that's the batsmen struggling though, you take your hats off to the Aussie bowlers standing up and being counted.
"Australia will want to win as many Test matches as they possibly can.
"There are two Tests left, and how great would it be to go home and say you drew the series."