As I've learnt, systems in the Universe always tend to increase their entropy since it provides a larger number of availiable microstates for energy to be distributed and is hence statisticaly preferred.
Also, colder objects tend to have a higher entropy since they want to break free of contraints applied due to the low temperature(such as being in a solid or liquid state) and hence "dissipate" energy and become more disordered.
But then, this would mean that systems always want to attain a higher state of disorder(which they could do by attaining a higher temperature, right?) which would imply that objects must never cool down since cooling down increases the stability at a molecular level and must be thermodynamically unfavourable, which is quite an obvious contradiction to what we observe in real life.
I am aware of the equations regarding change in entropy of universe, System and surroundings. My question is, shouldnt it intuitively unfavourable and hence never occur?