Arrows of Time and the Anisotropic Properties of CMB

The relation between the thermodynamical and cosmological arrows of time is usually viewed in the context of the initial conditions of the Universe. It is a necessary but not sufficient condition for ensuring the thermodynamical arrow. We point out that in the Friedmann-Robertson-Walker Universe with negative curvature, k=-1, there is the second necessary ingredient. It is based on the geodesic mixing - the dynamical instability of motion along null geodesics in hyperbolic space. Kolmogorov (algorithmic) complexity as a universal and experimentally measurable concept can be very useful in description of this chaotic behavior using the data on Cosmic Microwave Background radiation. The formulated {\it curvature anthropic principle} states the negative curvature as a necessary condition for the time asymmetric Universe with an observer.