The holographic principle says that quantum gravity equals a quantum field theory (without gravity) in one less dimension. The inspiration was Hawking's calculation that a black hole's entropy depends on its surface area rather than on its volume.
In the mathematical examples that have been worked out, the gravitational space has e.g. 4 large space-time dimensions and 7 small (compact) dimensions, and the field-theory dual has 3 large space-time dimensions. So the remaining dimensions are there too, but they are encoded differently.
However, those mathematical examples are for a negatively curved space (anti de Sitter space). Positively curved space (de Sitter space) may be closer to reality. In that case, time itself may be the extra holographic dimension, and the dual description is a timeless, purely three-dimensional space. But that dS/CFT duality has problems compared to AdS/CFT.
In both those examples, an entire space-time has a dual description in terms of a field theory on a space or space-time of one less dimension. There is probably a partial version in which you can project part of a space-time onto a surface inside it, like your example with the cosmological horizon, but I think it is only worked out for AdS/CFT.