Before the temperature gets that high, something strange happens...
On the Red Giant Branch, the core of the Sun is being compressed to
higher densities and temperatures, but it takes a temperature of 100 Million
Kelvins to ignite Helium fusion.
Thermal pressure is due to the random motions of particles at a given temperature. Higher temperature means faster random motions and higher pressures.
Pressure can also be generated by trying to pack particles in close together.
Q: How close can you pack electrons?
A: Heisenberg Uncertainty Principle
If pressure is supplied by electron degeneracy, it does not depend on temperature.
What happens when thermonuclear fusion begins in a stellar core supported by electron degeneracy?
For a few hours the fusion of helium into carbon in the core will produce a luminosity of 10 billion suns!
This energy is trapped in the core of the star, and the high pressure expands the core (but the envelope shrinks) until a stable burning is reached.
Life continues on the horizontal branch: a helium main sequence.
How long will the Sun burn Helium in the core?