  We consider Freund-Rubin-type compactifications which are described by
(p+q)-dimensional Einstein gravity with a positive cosmological constant and a
q-form flux. Using perturbative expansions of Kinoshita's ansatz for warped
dS_pxS^q and AdS_pxS^q spacetimes, we obtain analytical solutions describing
the warped branches and their respective phase spaces. These equations are
given by inhomogeneous Gegenbauer differential equations which can be solved by
the Green's function method. The requirement that the Green's functions are
regular provides constraints which determine the structure of the phase space
of the warped branches. We apply the perturbation results to calculate the
thermodynamic variables for the warped dS_pxS^q branch. In particular, the
first law of thermodynamics can be reproduced using this method.
