The problem is theoretical and practical... In theory, a cap would be the same for everyone... in practice, rich people would be able to avoid the cap, and the poor would still be capped.
I lived in a country, where we had odd- even- systems of driving a car due to gas shortages many years ago (basically if your licence plate ended with an odd number, you could drive on odd dates, and even-even). Rich people bought another car... poor people had transport problems.
We've come to a situation where rich people ride their private jets to eco-something conferences, tell Average Joes that they shouldn't eat meat and drive their cars, because in 20 years, the sea will rise for x meters anyway, and then those rich people take their private jets back to their newly bought beachfront properties.
So, when the law (which is written by the rich) changes... who will get fucked the most? WIll they really hurt their own? If they're "afraid" of climate change, why don't they change their behaviour first? And if they don't, why should average joe do that, when he already pollutes A LOT less?
And yet the wealthy, historically speaking, seem to have a real penchant for taking advantage of the poor in perverse ways.