Around time 33:40 min in this video, Prof. Paul Steinhardt says the following about gravity:

This argument is often brought forward, as in this case here, when inflationary cosmology is explained. The energy of the inflating universe comes from gravity or energy of the inflating universe is conserved since the energy of the inflation field is set off by the energy of gravity.

I also know from introductory QFT courses that gravity is special. When computing the vacuum energy, it is pointed out that for gravity not the potential energy difference but the total potential energy matters.

1. But where do I find this "bottomless feature" of gravity in the gravity and GR textbooks? Which chapter or formula in, say, Sean Carroll or in Bernard Schutz book explains it?

2. What makes gravity "bottomless"? Why does the same explanation does not apply for two opposite electric charges brought arbitrarly close to one another?

That is nothing special in gravity, you get the same potential for opposite electric charges. It is a mathematical artifact, however, he tries to apply a formula in a region where it is not valid. You cannot bringt two objects as close to each other as you like, and black holes have a finite energy as well.

I did not watch the video yet, but had the same reaction as was posted....

As far as I have learned, the only 'unique' thing about gravity is that it tilts light cones....
that and the related fact that it has not yet been included in the Standard Model because it stands alone from the other forces....

I'll be interested to hear what Steinhardt has to say. Thanks for the link.

edit: "When computing the vacuum energy, it is pointed out that for gravity not the potential energy difference but the total potential energy matters..."

Au contraire, there's a well-known theorem in GR, "For asymptotically flat gravitating systems
the total energy is well defined and must be non-negative."