  We discuss the interaction of a mobile quantum impurity with a Bose-Einstein
condensate of atoms at finite temperature. To describe the resulting Bose
polaron formation we extend the dynamical variational approach of
[Phys.Rev.Lett. 117, 11302 (2016)] to an initial thermal gas of Bogoliubov
phonons. We study the polaron formation after switching on the interaction,
e.g., by a radio-frequency (RF) pulse from a non-interacting to an interacting
state. To treat also the strongly-interacting regime, interaction terms beyond
the Fr\"ohlich model are taken into account. We calculate the real-time
impurity Green's function and discuss its temperature dependence. Furthermore,
we determine the RF absorption spectrum and find good agreement with recent
experimental observations. We predict temperature-induced shifts and a
substantial broadening of spectral lines. The analysis of the real-time Green's
function reveals a crossover to a linear temperature dependence of the thermal
decay rate of Bose polarons as unitary interactions are approached.
