ColorFunction and/or ColorScaling issue with ParametricPlot3D

Main thing to know is that for ColorData["TemperatureMap"][f[x]] you must have 0 < f < 1. What ColorFunctionScaling does - it rescales range x, and you need to rescale domain f.

I will consider a simplified case of your problem and you can generalize. Let's take t = 0.5 Stripping your function u of unneeded variables, you have:

u[y_,t_]=(-8*(2 - Pi)*Cos[(Pi*y)/4])/(E^((Pi^2*t)/20)*Pi^2) - 
 (8*(2 + 3*Pi)*Cos[(3*Pi*y)/4])/(9*E^((Pi^2*t)/5)*Pi^2) - 
 (8*(2 - 5*Pi)*Cos[(5*Pi*y)/4])/(25*E^((9*Pi^2*t)/20)*Pi^2)

Let's plot its original and re-scaled forms:

{min, max} = {NMinValue[#, y], NMaxValue[#, y]} &@{u[y, .5], 0 < y < 2}

{1.29488*10^-16, 0.789722}

Plot[{u[z, .5], (u[z, .5] - min)/(max - min)}, {z, 0, 2}, PlotRange -> All, Filling -> 0]

You can see the difference. Now, the correct cylinder for t = 0.5 is set up like this - and you can see exact correspondence to your rectangle pictures:

GraphicsRow[{

ParametricPlot3D[{Cos[theta], Sin[theta], z}, {theta, -Pi, Pi}, {z, 0, 2}, 
AxesLabel -> {x, y, z}, ColorFunctionScaling -> False, 
ColorFunction -> Function[{x, y, z, m, n}, 
             ColorData["TemperatureMap"][Rescale[u[z, .5], {min, max}]]], 
Mesh -> 8,  MeshFunctions -> 
     Function[{x, y, z, m, n}, Rescale[u[z, .5], {min, max}]]],

  ContourPlot[u[y, .5], {x, -Pi, Pi}, {y, 0, 2}, 
   ColorFunction -> ColorData["TemperatureMap"], Contours -> 8]}]

You can get the colors from the 2D contour plots and use them as the setting for MeshShading:

 tt = {0, .1, .3, .5, 1, 2, 3, 4}; 
 cps = Table[ContourPlot[u[x, y, t], {x, -a, a}, {y, 0, b}, 
   ColorFunction -> ColorData["TemperatureMap"], Contours -> 8],  {t, tt}];
 colors = Cases[cps[[#]], _RGBColor, Infinity] & /@ Range[8];

Then

Grid[Partition[Table[ParametricPlot3D[{Cos[theta], Sin[theta], rho}, 
       {theta, -Pi,  Pi}, {rho, 0, 2},
   Mesh -> 8,
   MeshFunctions -> {Function[{x, y, z, theta, rho}, u[x, z, tt[[i]]]]},
   MeshShading -> colors[[i]],
   ColorFunction -> Function[{x, y, z, theta, rho},
      ColorData["TemperatureMap"][u[x, z, tt[[i]]]]],
   AxesLabel -> {x, y, z}], {i, Range[8]}], 4]]

gives

I figured that since the question's been answered anyway, I'd illustrate the Texture[] alternative, which completely avoids the rigamarole of having to find the extrema of the coloring function:

a = π; b = 2;
f[x_, y_] := y;
u[x_, y_, t_] := -((8 E^(-((π^2 t)/20)) (2 - π) Cos[(π y)/4])/π^2) -
                 (8 E^(-((π^2 t)/5)) (2 + 3 π) Cos[(3 π y)/4])/(9 π^2) -
                 (8 E^(-((9 π^2 t)/20)) (2 - 5 π) Cos[(5 π y)/4])/(25 π^2);

tex = With[{t = 0.5},
           Image[ContourPlot[u[x, y, t], {x, -a, a}, {y, 0, b}, 
                             AspectRatio -> Automatic, ColorFunction -> "TemperatureMap", 
                             Contours -> 8, Frame -> None, ImagePadding -> None, 
                             PlotRange -> All, PlotRangePadding -> None], 
                 ImageResolution -> 256]];

ParametricPlot3D[{Cos[θ], Sin[θ], ρ}, {θ, -a, a}, {ρ, 0, b},
                 AxesLabel -> {x, y, z}, Lighting -> "Neutral", 
                 Mesh -> None, PlotStyle -> Texture[tex], 
                 ViewPoint -> {-2.3, 0.77, -2}, ViewVertical -> {-0.08, 1, -0.06}]

I'll leave the generalization as an exercise for the interested reader.