  We show that the homogeneously curved Friedmann-Robertson-Walker (FRW)
cosmology filled with a massless scalar field enjoys a Noether symmetry under
the 1d conformal group SL(2,$\mathbb{R}$). Concretely, the dynamics of the
system is invariant under M\"{o}bius transformations of the proper time
provided the scale factor transforms as a primary field. This extends previous
results obtained recently for the flat FRW case, the (A)dS cosmology and the
Kantowski-Sachs mini-superspace model. We compute the Noether charges
generating this hidden conformal symmetry and show that they form an
sl(2,$\mathbb{R}$) Lie algebra. The initial conditions of these evolving
constant of motion form the CVH algebra which fully encodes the dynamics of the
system. This structure reveals new dualities : on one hand, the cosmological
dynamics can be reformulated as a particle propagating on the AdS$_2$
hyperboloid and on the other hand, one can map the cosmological system onto
conformal mechanics. These reformulations of the cosmological dynamics open up
the possibility to import known results from these systems to classical and
quantum cosmology.
