What happens to the core of a high-mass star after it runs out of hydrogen?

It shrinks and heats up

Explanation:

When a star is in a moment of its life called Main Sequence, which means that it produces its energy by melting hydrogen nuclei into helium. This is because, during its birth, the gravitational pressure of its enormous mass caused the temperature to reach about 10 million degrees in its core, enough to initiate hydrogen fusion. For example, right now, the Sun is in a situation of hydrostatic equilibrium: on the one hand, the weight of its outer layers is gigantic, but on the other it is producing a large amount of radiation and the convection inside it is enormous. But, given the magnitude of the forces involved, it is a rather precarious balance.

Imagine a young Sun: a couple of billion years ago. At that time it has a lot of hydrogen and little helium. But, gradually, it is consuming hydrogen and gaining helium and, therefore (here is the key) getting denser. At that moment, the balance is broken - gravity wins. The Sun begins to compress, because the fusion is not able to compensate for its increase in density.

The conditions in the nucleus of the new Sun are more extreme: there is greater pressure and higher temperature (due to gravitational compression). The star begins to consume hydrogen faster, accelerating the fusion: at that time, it manages to compensate for the gravitational pressure and returns to equilibrium but is hotter. Since it began to fuse hydrogen, the Sun has increased its power by 40%, accelerating its end.