He says this is largely due to our sense of balance. "As you get older, your organ of balance tends to deteriorate and you're likely to feel more physically vulnerable."
Older adults also tend to have people who depend on them, and this can make them more troubled by the possibility of falling.
Worries such as these can intensify a phobia.
"As an adult, you're more aware of what you and your loved ones have to lose, and that's all part and parcel of vulnerability."
However, Professor Gournay adds that while acrophobia gets worse, other phobias tend to be less problematic with age.
"As you get older, you produce much less adrenaline - the fight or flight hormone - so a lot of phobias diminish." It's adrenaline that causes the racing heart and dizziness we experience when we're encountering something we're scared of, he explains.
"You might still have the fear, but you won't get the intense feelings you got when you were younger.
"Generally, phobias will probably improve with age, but if your phobia has anything to do with being vulnerable, such as heights or big crowds, it will probably get worse."
His theory is supported by a YouGov survey in 2014, which found 49 per cent of 18 to 24-year-olds were afraid of heights, compared with 61 per cent of 40 to 59-year-olds, and 64 per cent of over-60s.
Conversely, the younger adults were nearly twice as likely to be afraid of spiders (59 per cent) as the over-60s (32 per cent).
Phobias come in different forms: specific phobias related to things such as spiders, heights, blood or the dentist; or social phobias, where social situations such as speaking in public make you anxious.
Professor Gournay has noticed that health phobias - excessive worrying about becoming ill - are much more common in younger people. "Health phobias are very common indeed and, ironically, many of the people who present with them are young, fit and healthy. Perhaps as one gets older, one becomes more resigned to suffering illnesses, and eventually, to mortality."
The key to overcoming a phobia is confronting it - unless you do, it will get worse with time, not better, says Dr Blenkiron.
"Avoiding the situation obviously does help with the anxiety for a short term but, in the long term, it's a reinforcing behaviour and makes the whole thing worse."
The best treatment is exposure therapy. Here, the patient confronts the source of their phobia for increasing amounts of time.
"Instead of running away, you stick with it, so you suffer the anxiety in the short term, but in the long term, the phobia goes away," he says.
So there is good reason to try to overcome a phobia - some research even suggests the stress of living with one may shorten your life.
A 2012 study by Brigham and Women's Hospital in Boston, in the US, found that women with phobic anxiety - fears of crowds, heights or the outside - had shorter telomeres.
These cells are like caps that protect our DNA; shorter telomeres suggest premature ageing.
The good news is that the treatment is straightforward and inexpensive. "You don't need therapy or medication," says Professor Gournay. "Phobias can be very effectively treated with self-help.
"If you have a fear of heights and escalators, go to a John Lewis store and start with the first floor, then once that feels OK, try going to the second, until you can get to the fifth.
'I have a few patients who are - hopefully - doing this as I speak."
- Daily Mail
