Shouldn't $CO_2$ forcing depend on the surface temperature?

When I read about radiative forcing due to CO2 it says:

“Radiative forcing is a measure of the influence a factor has in
altering the balance of incoming and outgoing energy in the
Earth-atmosphere system and is an index of the importance of the
factor as a potential climate change mechanism. In this report
radiative forcing values are for changes relative to preindustrial
conditions defined at 1750 and are expressed in Watts per square meter
(W/m2).”

Then it goes on to give an equation of the forcing as a function of only the concentration and some constant:

$Delta F = 5.35 times ln {C over C_0}~mathrm{W}~mathrm{m}^{-2} , $

And reading about the greenhouse effect, wikipedia says:

Greenhouse gases absorb and emit some of the outgoing energy radiated
from Earth’s surface, causing that heat to be retained in the lower
atmosphere.[38]

If the CO2 is reflecting back IR radiation that was emitted from the surface (the amount of which obviously depends on the temperature), shouldn’t the forcing depend on surface temperature? Ie, it should be higher at the equator than near the poles?

To use an extreme example, I would expect the CO2 forcing for a planet at 0 K to also be 0 W/m^2, since there would be no IR emitted by the surface for it to absorb.

If that is correct, are there any papers about greenhouse gas forcings as a function of temperature/latitude? I do not mean due to latitude-dependent concentration of CO2, just due to the temperature of the surface.

EDIT:

I did some more research and found some data that shows that IR emissions for earth by latitude ranges from 94-276 W/m^2. So if the forcing is assumed constant, even just for our planet (not considering the rest of the solar system or exoplanets), it is equivalent to assuming that a given concentration of CO2 radiates back the same amount of energy over a ~3-fold difference in input energy.

Edit 2:

I asked this in a comment and think it may be a more direct question than the original:

If a 0.2% CO2 atmosphere receives 300 W/m^2 of IR from the surface, will it radiate
back towards the surface the same amount as a 0.2% CO2 atmosphere that
receives 100 W/m^2 of IR from the surface?

Edit 3:

I found a derivation for that 5.35 value. Apparently it is:
$F = f_a*f*F_{g►a}*(1 – T_d) = 0.6*0.4256*390*0.05371 = 5.35$

where:

• $F$ is the $CO_2$ greenhouse flux density in $W/m^2$ ($F_{a►g}$)

• $f_a$ is the fraction of flux returned downward to the Earth,
  absorbed or re-emitted by $CO_2$

• $f$ is the Planck Blackbody Fraction (The fraction of Earth’s heat
  radiation in the spectral interval over which $CO_2$ absorption is
  significant)

• $F_{g►a}$ is the total flux density emitted by the Earth’s surface
  ($sigma T^4$)

• $T_d$ is the diffuse transmittance

• $(1-T_d)$ is the fractional absorption

The value of interest would then be:
$F_{g►a} = sigma T^4 = 390,Wm^{-2}$

We know the Stefan-Boltzman constant is $sigma = 5.67 x 10^{-8},Wm^{-2}K^{-4}$. So to get a value of $390,Wm^{-2}$ they must have used $T = 288,K$, which makes sense since that is a widely cited value for the average temperature of the earth.

The temperature on earth has been found to vary between $183.95-367.05,K$, which corresponds to values of $F_{g►a}$ between $65-1029,Wm^{-2}$. The forcing will then range between:

$Delta F = 0.89 times ln {C over C_0}~mathrm{W}~mathrm{m}^{-2} , $

$Delta F = 14.11 times ln {C over C_0}~mathrm{W}~mathrm{m}^{-2} , $

So that is about a factor of 16 difference in forcing due to different surface temperatures. Correct?

Edit 4:

I should also note that the forcing due to the average temperature is not equal to the average forcing of multiple temperatures:
$$sigmafrac{1}{n}sum_{i=0}^n T_i^4 neq sigma(frac{1}{n}sum_{i=0}^n T_i)^4$$

You can easily prove it to yourself by plugging in two temperatures 298 and 278 K that average to 288 K. The $LHS approx 393$ and $RHS approx 390$. Can anyone cite something that tells us which calculation is used in GCMs?

Edit 5:

At least in the GISS model E v2.1.1 GCM it appears the same forcing based on an average temperature of 288 K is applied everywhere. From model/RAD_DRV.f:

#ifdef HEALY_LM_DIAGS
      FCO2=FULGAS(2)*CO2R
      FN2O=FULGAS(6)*N2OR
      FCH4=FULGAS(7)*CH4R
c
c      write(6,*) 'RJH: GHG: CONC=',
c     * FCO2,FN2O,FCH4
      ghg_totforc=5.35d0*log(FCO2/CO2I)
     *  +0.036d0*(sqrt(FCH4)-sqrt(CH4I))
     *  -(Fe(FCH4,N2OI)-Fe(CH4I,N2OI))
     *  + 0.12d0*(sqrt(FN2O)-sqrt(N2OI))
     *  -(Fe(CH4I,FN2O)-Fe(CH4I,N2OI))
c      write(6,*) 'RJH: GHG: FORC=',ghg_totforc
#endif

Notice the 5.35d0*log(FCO2/CO2I) at line 2156. I don’t know fortran though, so perhaps this code is doing something else.

Edit 6:

Looking further at GISS model E2.1.1, it appears I have answered the title question. Below is a extract of what they do to calculate the impact of CO2 on the downward solar flux within a given spectral interval for a single atmospheric layer at a given solar zenith angle:

C In model/RADIATION.f
C ULGAS  = Array of gas concentrations by layer (CO2 is column 2)
C N      = Atmospheric layer
C TLN    = Temperature at the middle of layer N
C PLN    = Pressure at the middle of layer N
C ULN    = Concentration of CO2 in layer N
C TAU    = Spectral Optical Thickness 
C K      = Spectral Interval (eg, K = 1 corresponds to 3000-3400 nm)
C DKSO   = Fraction of solar flux in interval K (eg, .01 for K = 1)
C S0COSZ = Solar constant (defined as 1366 W/m^2) times cosine of the solar zenith angle
C DKS0X  = Solar flux in spectral interval K at zenith angle Z
C NL     = Top atmospheric layer
C SRDFLB = Vector of total solar downward flux at bottom edge of each layer

C Line 5586:
 DKS0X=DKS0(K)*S0COSZ


C Line 5605:
C     Select parameterized k-distribution gas absorption by H2O, O2, CO2

C Line 5678:
      CASE (9)
C--------K=4-------CO2       DS0=.002
      ULN=ULGAS(N,2)
      TERMA=(50.73-.03155*TLN-PLN*(.5543+.00091*TLN))*(1.-.1004*PLN)
      TERMB=(1.+.006468*ULN)*(1.+PLN*(49.51+.8285*PLN))
      TAU  =(TERMA/TERMB)*ULN
      IF(PLN < 175.0) TAU=(.00018*PLN+0.00001)*ULN

      CASE (10)
C--------K=3-------CO2       DS0=.003
      ULN=ULGAS(N,2)
      TERMA=(1.+.01319*TLN)*(PLN*(.008001*ULN+.4589E-03)-.8396*ULN)
      TERMB=ULN*(PLN+295.7+1.967*ULN)+.15126*PLN
      TAU  =(TERMA/TERMB)*ULN

      CASE (11)
C--------K=2-------CO2       DS0=.003
      ULN=ULGAS(N,2)
      TERMA=(1.+.02257*TLN)*(PLN*(.002295*ULN-.5489E-04)-.7571*ULN)
      TERMB=ULN*(PLN+803.9+2.477*ULN)-.09899*PLN
      TAU  =(TERMA/TERMB)*ULN

C Line 5823:
 XANX   = EXP(-TAU*SECZ)
 DX     = 0

C Line 5844:
 SRDFLB(NL)=SRDFLB(NL)+DKS0X*(XANX+DX)

By seeing how they use TLN (Temperature of layer N) to determine TAU (optical depth), we see the answer is that the forcing due to CO2 is a function of the temperature of that atmospheric layer (which would be a function of the surface temperature). It is also a function of the pressure, and concentration of the gas in that layer.

I isn’t clear to me what all those magic numbers are in the calculation for TAU, nor what they are doing with XANX = EXP(-TAU*SECZ), but definitely see that the forcing is a function of the temperature.

The “simplified” calculation for the CO2 forcing appears to be worse than useless… it is not even the correct average. So I don’t know what that was doing in their code, hopefully it was not used to tune the model.