It is now for a long period very warm for Belgium, normally we don't have such warm weather for such a long time, could that be a cause?

I'm also from Belgium and had already considered a similar thing for our cats. One of them is incessantly ravenous and never refuses food (he used to live on the street so he values any meal he gets), but he's refused to eat for two days now even though he still asks for food.

One thing you should consider is that heat facilitates smell. The warmer food is, the more of a smell it releases. Maybe the cat food you're giving was borderline okay in the past, but it currently smells too bad for your cat to want to eat it. (Our cat seems to specifically decide to not eat the food after smelling and investigating the food).

I initially considered if the cat simply feels unwell due to the heat, but it doesn't make sense that they hound me for food then outright refuse it. He's clearly hungry, and intends to eat the bowl I'm putting down, but then suddenly refuses after inspecting it.

Whenever your cat refuses to eat, if you want to make sure that it's not a medical issue, give them their favorite food (a treat, human food they love, ...) and see if they eat that. If they don't even seem interested or are unable to actually keep it in, it's time to go to the vet.

I have a cat that is at an older age (we don't know her age, we only know she is older than 12 years). She always liked to eat and would eat anything you give her. The last two months this changed. She doesn't want to eat her food but asks for it. Every time we give her food, she smells it and starts to ask for food again without touching the food.

However, there are also some different considerations:

Older cats are more set in their ways. They're more likely to have standards and might be very inflexible about them. However, since you mention that she was easy going a few months ago, this may not be the case now.

For reference, our 3 cats were all rescues from the street. They were initially happy to eat any meal provided. Because I used to have an older cat who was very particular about the food she'd eat, I've intentionally tried to prevent a standard from forming with our 3 cats. I never buy them the same brand food in a row, and I switch between most brands that seem good enough.

While I can clearly see their preferences, I am intentionally refusing to cater to them because you're going to eventually pain yourself in a corner with the food options that your cat will accept.

Have you replaced meals when your cat didn't eat? Because that can implicitly teach your cat that it can ask you for a replacement meal. You need to be hardheaded here. Refuse to replace the meal. You can't make your cat eat the food, but you can get them to choose between having a mediocre meal or not having a meal at all. It's up to them to decide.
Similar to children, I give my cats no treats unless they're reasonably finished their meal, so they don't get a substitute snack.

Safety warning: don't be hardheaded enough to ignore a cat who's willing to starve itself. Medical needs come before training.

Are you using plastic bowls? Plastic bowls often have microscopic pores, which can over time get clogged with food remains, which start to give off a bad smell after a while (and no amount of normal washing will clear this). It's possible that the smell has accumulated on the bowl enough that it ruins your cat's appetite. Try feeding them from a new container and see if that changes anything.
In regards to a long term solution, favor metal bowls over plastic ones. They accumulate less food remains because they're not as porous.