Why is cooling much harder than heating?
It is because of the second law of thermodynamics. There are many irreversible processes that can be used to heat something. It is the natural flow of things because entropy will increase in isolated systems, and much of the internal energy of objects can be dissipated as heat (and this heat used to heat something that is colder). However, to cool something you need to perform work in order to decrease the entropy of the subsystem. There is a maximum efficiency for work to be used to cool something by removing heat from one source at lower temperature and move it to a source at a larger temperature. So both processes, heating and cooling are not symmetrical in our universe with a thermodynamic arrow of time.
You could blame the laws of thermodynamics and say that cooling is much harder in our universe because of them. However, since we're in a dark energy-dominated universe that's expanding and cooling, it seems as though cooling is generally easier for the universe on the largest scales.
Even on smaller scales, cooling is usually easier (I've of course determined ease by observing which one happens more often). The cores of planets will cool and harden over time, stars use up their fuel and cool as they die. Cooling by far is the easier process. Even when the universe reaches a heat death (assuming it ever does), the expansion will continue to cool it to lower temperatures. So cooling is definitely easier for the universe.
So why isn't cooling easier for us? Well the answer could be due to thermodynamics. It could be that on short time and distance scales, it is easier to heat than to cool, but let me present a more anthropological reason.
Throughout human history, we have always strove to perform tasks. We want to build buildings, grow crops, light our homes. All these tasks require us to expend or use energy. As such, we have invented brilliant systems and processes of generating easy-to-use energy and channeling it where we need it to be. We have become very adept at taking energy from a few common sources and dumping it into wherever or whatever we want. And, as we all know, putting energy into an object is much the same as increasing its temperature. So, for us, increasing the temperature of something is no problem. It's generally what we do. However, to decrease the temperature of an object, you need to remove energy from it. Now with the exception of those few specific sources, we aren't very adept at taking energy out of something. That has never really been as necessary in history because usually when you remove energy, it makes it harder to do any tasks. What would the point be of making it harder to perform tasks? As such, we rely mostly on natural processes to remove the energy from systems. But unlike the processes we invented, natural processes usually try to bring temperatures to thermal equilibrium. Sure, we invented refrigeration and we found a few endothermic reactions to exploit. But when it comes down to it, we're much more interested in putting energy into things (computers, lights, heating systems, anything requiring electricity) than taking it out.
It could simply be the case that while the universe finds cooling to be easier, we have put a lot more effort into figuring out how to heat things and so that is easier for us.
Let me offer a different persepective on this. Cooling is not universally harder than heating. To demonstrate this, consider the following:
Suppose you have two 1 kg copper blocks, one at 200 K, and one at 400 K. Put them into direct contact, and seal them in a perfectly-insulated (no heat transfer from the walls) empty box. The 400 K block will then cool to 300 K just as readily as the 200 K block warms to 300 K. [N.B.: The 300 K final temperature is based on the simplifying assumption that the heat capacity of copper is temperature-independent between 200 K and 400 K.]
But, you might protest, aren't there many instances in which cooling is more difficult—e.g., cooling vs. heating a home? The answer is yes. But then what distinguishes the example I've offered from those that concern you? The difference is that you are specifically referring to cases in which potential energy (PE) (typically electrical energy) is used for heating or cooling.
Given this, I would suggest your question be refined as follows: Why is it harder to use PE to cool than to heat? Now I can give you an answer:
PE can be converted to thermal energy with no losses, thus achieving heating directly with 100% efficiency. E.g., I can convert electrical energy entirely to thermal energy using a resistor. [This ignores losses in getting the PE into your system, e.g., resistive losses in electrical transmission.]
However, PE cannot be directly converted into the removal of thermal energy. Instead, to use PE to cool requires running some form of heat engine. And, by the 2nd law of thermodynamics, even a perfect heat engine cannot be 100% efficient. Furthermore, all real-world heat engines operate dissipatively, and thus will be even less efficient than a perfect heat engine.
So, in summary, if you are talking about the conversion of PE to heating vs. cooling, the difference is that the former can be done directly, with 100% efficiency, while the latter always requires going through some form of heat engine, with significant attendant losses (as well as [with some exceptions, such as a Coolgardie safe] significantly greater engineering complexity).