Correct, that's a key fact about the underlying effects in question, of which Hawking radiation is just a consequence: different observers see different particles in the vacuum.
In 1976, Bill Unruh published "Notes on black-hole evaporation"[1], in which he showed that "an accelerated detector even in flat spacetime will detect particles in the vacuum" - now known as the Unruh effect This means that an observer in an accelerated reference frame will observe particles in the vacuum where an inertial observer will observe none. The presence of certain particles - the ones we call Hawking radiation in the context of a black hole - is a relative phenomenon. This is known as the Unruh effect. The equation for the Hawking temperature is essentially the same as the equation for the Unruh temperature, where the acceleration value is the acceleration due to gravity of the black hole.
Then in 1977, Gibbons and Hawking published "Cosmological event horizons, thermodynamics, and particle creation"[2], which showed that "the close connection between event horizons and thermodynamics which has been found in the case of black holes can be extended to cosmological models with a repulsive cosmological constant" and that "An observer with a particle detector will indeed observe a background of thermal radiation coming apparently from the cosmological event horizon." This is known as the Gibbons-Hawking effect.
There's a fairly complex relationship between the two effects which I won't try to describe, but if you're interested then [3] discusses it. The abstract itself gives some sense of the connection.
In 1976, Bill Unruh published "Notes on black-hole evaporation"[1], in which he showed that "an accelerated detector even in flat spacetime will detect particles in the vacuum" - now known as the Unruh effect This means that an observer in an accelerated reference frame will observe particles in the vacuum where an inertial observer will observe none. The presence of certain particles - the ones we call Hawking radiation in the context of a black hole - is a relative phenomenon. This is known as the Unruh effect. The equation for the Hawking temperature is essentially the same as the equation for the Unruh temperature, where the acceleration value is the acceleration due to gravity of the black hole.
Then in 1977, Gibbons and Hawking published "Cosmological event horizons, thermodynamics, and particle creation"[2], which showed that "the close connection between event horizons and thermodynamics which has been found in the case of black holes can be extended to cosmological models with a repulsive cosmological constant" and that "An observer with a particle detector will indeed observe a background of thermal radiation coming apparently from the cosmological event horizon." This is known as the Gibbons-Hawking effect.
There's a fairly complex relationship between the two effects which I won't try to describe, but if you're interested then [3] discusses it. The abstract itself gives some sense of the connection.
[1] https://journals.aps.org/prd/abstract/10.1103/PhysRevD.14.87...
[2] https://journals.aps.org/prd/abstract/10.1103/PhysRevD.15.27...
[3] https://arxiv.org/abs/2211.14747