How much did the universe expand since recombination time? 
Assuming adiabatic expansion, and a temperature at recombination time of $2700^{\circ} C$, how much has the universe expanded if the temperature today is $2.7K$?

My idea is to use
$\frac{dQ}{dt}=A\sigma T^4$
where $\frac{dQ}{dt}=0$, A is the blackbody surface area, and T is the temperature. The way I did it was since the expansion is adiabatic (subscript $0$ for recombination time and $f$ for today):
$A_0\sigma T_0^4=A_f\sigma T_f^4$
and solving for $\frac{A_f}{A_0}$:
$\frac{A_f}{A_0}=\frac{T_0^4}{T_f^4}=\frac{(2973K)^4}{(2.7K)^4}=1.47\times10^{12}$.
So the surface area of the universe is $1.47\times 10^{12}$ larger now than at recombination time. Is this correct?
 A: The temperature $T$ of the cosmic background radiation is given by the energy of the photons, which redshift proportionally to the expansion of the Universe. That is,
$$
a T = a_0 T_0,
$$
where $a$ is the scale factor, and subscript $0$ denote values today. Hence, when $T$ drops to a fraction $T_0 / T = 2.7\,\mathrm{K} \,/\, 3000\,\mathrm{K} \simeq 1/1100$, that means that the Universe has expanded by a factor $a_0/a = 1100$.
Usually we define the scale factor today to be unity, and just say $a$ was $1/1100 \simeq 9\times10^{-4}$ at that time. The redshift $z$ of the photons received from that time is then $z+1=1/a\simeq1100$.
This expansion is in all directions. So distances between galaxies (that are not so close that they are gravitationally bound) increase by a factor $a^{-1}$, the surface area of the observable Universe increases by a factor $a^{-2} \simeq 1.2\times10^6$, and the volume of the Universe increases by a factor $a^{-3} \simeq 1.3\times10^9$.
Note that when you talk about how much the Universe has expanded, it is common to quote the "linear" expansion, i.e. the scale factor $a$, although in principle the volume factor $a^3$ might strictly be a more appropriate term. In contrast, I've never seen anybody talk about how much the surface area of the observable Universe has expanded; $a^2$ doesn't really bear any physical significance, but $a$ and $a^3$ does, as they describe the change in distances between objects, and volumes and — in particular — densities of objects, respectively.
