  We study Gaussian quantum steering and its asymmetry in the background of a
Schwarzschild black hole. We present a Gaussian channel description of quantum
state evolution under the influence of the Hawking radiation. We find that
thermal noise introduced by Hawking effect will destroy the steerability
between an inertial observer Alice and an accelerated observer Bob who hovers
outside the event horizon, while it generates steerability between Bob and a
hypothetical observer anti-Bob inside the event horizon. Unlike entanglement
behaviors in curved spacetime, here the steering from Alice to Bob suffers from
a "sudden death" and the steering from anti-Bob to Bob experiences a "sudden
birth" with increasing Hawking temperature. We also find that the Gaussian
steering is always asymmetric and the maximum steering asymmetry cannot exceed
$\ln 2$, which means the state never evolves to an extremal asymmetry state.
Furthermore, we obtain the parameter settings that maximize steering asymmetry
and find that (i) $s=arccosh(\frac{\cosh^2 r}{1-\sinh^2r})$ is the critical
point of steering asymmetry, and (ii) the attainment of maximal steering
asymmetry indicates the transition between one-way steerability and both-way
steerability for the two-mode Gaussian state under the influence of Hawking
radiation.
