  We study the spectral action approach to higher derivative gravity. The work
focuses on the classical aspects. We derive the complete and simplified form of
the purely gravitational action up to the 6-derivative terms. We also derive
the equivalent forms of the action, which might prove useful in different
applications, namely Riemann- and Weyl-dominated representations. The spectral
action provides a rather rigid structure of the higher derivative part of the
theory. We discuss the possible consequences of this rigidness. As one of the
applications, we check whether the conformal backgrounds are preferred in some
way on the classical level, with the conclusion that at this level, there is no
obvious reason for such a preference, the space $S^1 \times S^3$ studied in
earlier works being a special case. Some other possible properties of the
higher derivative gravity given by the spectral action are briefly discussed.
