Since your code, doesn't work, I wrote my own simplified version:

    g[x_,t_] = NDSolveValue[{
      D[f[x, t], t] == D[f[x, t], {x, 2}]
      , f[x, 0] == Sin[x]
      , f[0, t] == 0, f[π, t] == 0
     }
     , f[x, t]
     , {x, 0, 1}, {t, 0, 5}];

Note that I am using `NDSolveValue` to spit out the function automatically. You can then directly integrate the function:

    integratedG[t_] = Integrate[g[x, t], {x, 0, 1}];

and the result is another interpolating function that you can then plot:

[![enter image description here][1]][1]

The reason that this works is that behind the scenes, an `InterpolatingFunction` consists of a bunch of polynomials, which are obviously easy to analytically integrate, and *Mathematica* does that automatically.

Alternatively, if you don't need the function itself, you can build this directly into the call to `NDSolveValue`:

    integratedG[t_] = NDSolveValue[{
       D[f[x, t], t] == D[f[x, t], {x, 2}]
       , f[x, 0] == Sin[x]
       , f[0, t] == 0
       , f[π, t] == 0}
      , Integrate[f[x0, t], {x0, 0, 1}]
      , {x, 0, 1}, {t, 0, 5}];

  [1]: https://i.sstatic.net/kf8s4.png