I do not understand well what you mean by Killing horizon here. A Killing horizon in a 4-dimensional spacetime is, by definition, a null 3-surface made of integral lines of a Killing field defined in a neighbourhood of that surface, which becomes null exactly on that surface. As is known, the surface gravity turns out to be always constant along **each** integral line (which can be proved to be geodesic).

I guess that actually you are wondering if the surface gravity must be constant also **changing** the integral curve of the Killing field while remaining on the horizon.

It is possible to prove that, whenever a spacetime admitting a Killing horizon satisfies Einstein equations and the dominant energy condition is verified, the surface gravity must be constant on

the whole horizon.

However a result, originally obtained by Carter, states that the surface gravity of a Killing horizon must be constant if (i) the Killing field is **static** or (ii) there is an additional Killing

field and the two Killing fields are 2-surface orthogonal.

All that can be found in Wald's textbook on general relativity. Now I cannot check the precise page, but I am sure on that.