  We advance a new viewpoint on the connection between the the thermodynamical
and cosmological arrows of time, which can be traced via the properties of
Cosmic Microwave Background (CMB) radiation. We show that in the
Friedmann-Robertson-Walker Universe with negative curvature there is a
necessary ingredient for the existence of the thermodynamical arrow of time. It
is based on the dynamical instability of motion along null geodesics in a
hyperbolic space. Together with special (de-correlated) initial conditions,
this mechanism is sufficient for the thermodynamical arrow, whereas the special
initial conditions alone are able to generate only a pre-arrow of time. Since
the negatively curved space will expand forever, this provides a direct
connection between the thermodynamical and cosmological arrows of time. The
structural stability of the geodesic flows on hyperbolic spaces and hence the
robustness of the proposed mechanism is especially stressed. We then point out
that the main relations of equilibrium statistical thermodynamics (including
the second law) do not necessarily depend on any arrow of time. Finally we
formulate a {\it curvature anthropic principle}, which stipulates the negative
curvature as a necessary condition for the time asymmetric Universe with an
observer. CMB has to carry the signature of this principle as well.
