I have some trouble asking this question, so I will try a roundabout approach to explain what I mean.
If (outside of Earth atmosphere) one looks at the sun from different distances, the light coming from any direction intersecting the Sun's surface has a BB spectrum which is always the same, that of the surface temperature of the Sun. Looking in any other direction it is a BB spectrum at 2.7°K. So the spectrum at any place is "thermal by parts". Far from the Sun the high temperature spectrum is within a smaller solid angle than closer but however close one gets (while still just outside the Sun) the "high temperature" thermal spectrum never concerns more than a 2$\pi$ solid angle.
Now consider a black hole emitting Hawking's radiation. Let us assume it is small enough its Hawking temperature is higher than 2.7°K. So if I understand correctly, if I am stationary outside the black hole (not free falling into it, which means I am effectively in an accelerated frame) I will see BB radiation at the Hawking temperature, but only in the solid angle where the line of sight intersects the black hole horizon. For other directions, I only get 2.7°K. Or am I already mistaken ? The closer I am the larger this solid angle. There is the added complication that when I get closer the spectrum is blue-shifted, but anyway the solid angle where I see BB temperature will never be more than 2$\pi$. Indeed if I cross the horizon, then I cannot be stationary anymore, and the logic of Hawking's radiation cannot apply in the same way.
So here is my question. The Hawking radiation is often compared to the Unruh effect, because in an accelerated frame there is also a horizon, and the expression of the Unruh temperature and Hawking temperature are the same.
But from what I read, it always seems that the Unruh temperature concerns a BB radiation in all directions, over a full 4$\pi$ solid angle. Unless I am mistaken.
So why is the Unruh temperature BB in the full 4$\pi$ solid angle rather than just in a "half-space" 2$\pi$ solid angle like the Hawking radiation at the limit where I get very close to the horizon ?