Electromagnetism is due to exchange of photons. Gravity is due to an exchange of gravitons.
Large numbers of photons have entropy. So do large numbers of gravitons.
Thus the electromagnetic field has entropy. Therefore, so does gravity and space-time.
Thus electromagnetism is entropic. Therefore, so is gravity and space-time.
There is nothing really new in all this. Nor is there anything to be questioned about it.
Why do people like Lubos Motl and Peter Woit get so upset about entropic gravity? This is a good case for studying prejudices.
Motl is upset because the argument makes it clear that an explanation of gravity with gravitons does not need superstrings. And he is upset because the argument makes clear that gravity is a purely three-dimensional effect. This does not fit his prejudice that the world is 11-dimensional.
Woit is upset because the argument is so simple. Woit, despite his fight against many prejudices, also promotes some of his own prejudices. And one that he promotes is: nature is complicated and needs complicated mathematics for its description.
We see that the most disparate scientists can criticize obviously true statements because of their prejudices. For both of them, prejudices count more than facts.