Refrigeration question - apple dessert with high sugar content

Please let me know if this is not the right newsgroup for this question.

My mother-in-law says that due to its high sugar content an apple crisp
dessert does not need to be refrigerated for the first day after cooking.
She says that if she made it just after breakfast and serves it just after
supper it will be safe to eat that evening and it needn't be refrigerated
until after it is served after supper.

I said any cooked food will start harboring bacteria just as soon as it
begins to cool, so she should have refrigerated it shortly after breakfast
and after supper she should only have warmed up what she anticipated we
would eat and kept the rest refrigerated.

Well she got all huffy and picked up her apple crisp and went home. My
thinking was, "Good riddens," but my wife asked me if I was absolutely sure
that it needed to be refrigerated and if there was any truth to her mom's
comment about the apple crisp's high sugar content making no refrigeration
necessary the first day after cooking?

I said, "No, I'm not sure and maybe she really knows what she was talking
about." My wife then said, "Well, you better find out because if you are
wrong and she is right then you owe mom an apology." I said, "Okay, I'll get
on the web and find out."

I looked and looked and the best I could find was that apple crisp keeps for
three days WITH refrigeration. I could not find anything which specified how
long it could go without refrigeration on the day it was prepared.

I am hoping someone here knows or can tell me where to look. I want to know
who was right and who was wrong.

Her mom's explanation is she grew up is the country where almost nobody
refrigerated anything. I mentioned this to my wife and said to her, "Do you
have any idea why you get stomach aches about five times more often than I
do?" She said, "No," and I replied it is because you eat about five times
more of your mom's cooking than I do and she never refrigerates anything
until it is time to go to bed." I said this isn't necessarily bad because
medical researchers are now saying that because people's bowels are less
active than they were fifty or sixty years ago people are gettin a lot more
colon cancer than they once got many years ago, so these frequent stomach
aches are not nessarily a bad thing.

Thnks for a your opinion on this subject.

Bob