  We theoretically search for dynamical cross-correlated responses of
three-dimensional topological superconductors and superfluids. It has been
suggested that a gravitational topological term, which is analogous to the
theta term in topological insulators, can be derived in three-dimensional
time-reversal invariant topological superconductors and superfluids, and that
the dynamical gravitational axion field can be realized by the fluctuation of
the relative phase, i.e., by the Leggett mode between topological $p$-wave
pairing and conventional $s$-wave pairing. In the presence of the dynamical
gravitational axion field, we propose the emergence of the "chiral
gravitomagnetic effect", a thermal current generation by gravitomagnetic
fields, i.e., by mechanical rotations. This effect can be regarded as a thermal
counterpart of the chiral magnetic effect which has been studied mainly in Weyl
semimetals. We also show the occurrence of the anomalous thermal Hall effect in
the bulk. We discuss a possible application of our study to the thermal
responses of Weyl superconductors.
