How is the filter response affected by the sampling rate?
It is not changing the filter cut-off frequency.
However, the x axis is from 0 to fs/2. As you vary the sampling frequency the x axis limits effectively change and the cut off frequency location within the graph changes.

It doesn't change the transfer function ?

It does. The filter coefficients as well as the response. Below is my result in Octave software for a first order low pass filter
   1
 -----
 s + 1


c2d(..., 0.1, 'tustin')

 0.04762 z + 0.04762
 -------------------
     z - 0.9048

Sampling time: 0.1 s



c2d(..., 0.05, 'tustin')

 0.02439 z + 0.02439
 -------------------
     z - 0.9512
Sampling time: 0.05 s


Even though the cut off frequency is the same, the response is different especially at high frequencies. As mentioned in the other answer,the warping effects which are significant near the Nyquist frequency occur differently for filters of different sampling frequency.