Heavy Cav Archers have 13 techs to be researched and 14 for civs with bonus techs like the Turks lets look at a generic civ with all techs available like the mongols.
The 13 techs together cost 3300 food, 2225 gold and 550 wood plus the cost for a Blacksmith, Barracks, Stables, Archery range and University. The worst part on top of the enormous price and the number of buildings required is the units cost wood and gold but the techs to upgrade them costs mostly food and gold. So not only do you need a great wood and gold eco but you need a lot on food as well. When producing other units you can mostly just focus on the resources you need to make them with Cav archers you need to basically gather all 3 resources. Since you need over 3000 food while also paying food for villagers and Imperial age upgrades.
Comparing the Cav archers cost to Paladin:
Paladins have 10 techs, 3 less than Cav archers. The cost of fully upgraded Paladin is 3295 food and 1795 gold plus the cost of a blacksmith, barracks and stables, Thats already two buildings less than Cav archers making them cheaper and easier to upgrade. So thats a total resource cost of 6075 for Heavy Cav archers vs 5090 for Paladin, a difference of 985 purely from the upgrades themselves and then on top of that you have the added cost of having to build another extra 2 buildings and researching an extra 3 more techs.
So, can you justify the cost of Heavy Cav Archers or do they need a buff?