I am using libav 9.6, installed via Homebrew.
$ avprobe -version
avprobe version 9.6, Copyright (c) 2007-2013 the Libav developers
  built on Jun  8 2013 02:44:19 with Apple LLVM version 4.2 (clang-425.0.24) (based on LLVM 3.2svn)
avprobe 9.6
libavutil     52.  3. 0 / 52.  3. 0
libavcodec    54. 35. 0 / 54. 35. 0
libavformat   54. 20. 3 / 54. 20. 3
libavdevice   53.  2. 0 / 53.  2. 0
libavfilter    3.  3. 0 /  3.  3. 0
libavresample  1.  0. 1 /  1.  0. 1
libswscale     2.  1. 1 /  2.  1. 1
Even though the sampling rate is displayed in the stdout in the command line output, the -show_format option doesn't surface the sampling rate information for the audio file at all.
Here is the BASH terminal output:
$ avprobe  -v verbose -show_format -of json  sample.gsm
avprobe version 9.6, Copyright (c) 2007-2013 the Libav developers
  built on Jun  8 2013 02:44:19 with Apple LLVM version 4.2 (clang-425.0.24)
(based on LLVM 3.2svn)
  configuration: --prefix=/usr/local/Cellar/libav/9.6 --enable-shared
--enable-pthreads --enable-gpl --enable-version3 --enable-nonfree
--enable-hardcoded-tables --enable-avresample --enable-vda --enable-gnutls
--enable-runtime-cpudetect --disable-indev=jack --cc=cc --host-cflags=
--host-ldflags= --enable-libx264 --enable-libfaac --enable-libmp3lame
--enable-libxvid --enable-avplay
  libavutil     52.  3. 0 / 52.  3. 0
  libavcodec    54. 35. 0 / 54. 35. 0
  libavformat   54. 20. 3 / 54. 20. 3
  libavdevice   53.  2. 0 / 53.  2. 0
  libavfilter    3.  3. 0 /  3.  3. 0
  libavresample  1.  0. 1 /  1.  0. 1
  libswscale     2.  1. 1 /  2.  1. 1
[gsm @ 0x7f8012806600] Estimating duration from bitrate, this may be inaccurate
Input #0, gsm, from 'sample.gsm':
  Duration: 00:03:52.32, start: 0.000000, bitrate: 13 kb/s
    Stream #0.0: Audio: gsm, 8000 Hz, mono, s16, 13 kb/s
{  "format" : {
    "filename" : "sample.gsm",
    "nb_streams" : 1,
    "format_name" : "gsm",
    "format_long_name" : "raw GSM",
    "start_time" : "0.000000",
    "duration" : "232.320000",
    "size" : "383328.000000",
    "bit_rate" : "13200.000000"
  }}
And the python code example:
>>> filename = 'sample.gsm'
>>> result = subprocess.check_output(['avprobe', '-show_format', '-of',
'json', filename])
avprobe version 9.6, Copyright (c) 2007-2013 the Libav developers
  built on Jun  8 2013 02:44:19 with Apple LLVM version 4.2
(clang-425.0.24) (based on LLVM 3.2svn)
[gsm @ 0x7fe0b1806600] Estimating duration from bitrate, this may be
inaccurate
Input #0, gsm, from 'sample.gsm':
  Duration: 00:03:52.32, start: 0.000000, bitrate: 13 kb/s
    Stream #0.0: Audio: gsm, 8000 Hz, mono, s16, 13 kb/s
>>> print result
{  "format" : {
    "filename" : "sample.gsm",
    "nb_streams" : 1,
    "format_name" : "gsm",
    "format_long_name" : "raw GSM",
    "start_time" : "0.000000",
    "duration" : "232.320000",
    "size" : "383328.000000",
    "bit_rate" : "13200.000000"
}}
So I am aware that sampling rate could be a stream specific display to be shown in -show_format option results. But there isn't any other options to detect the sampling rate on a specific audio stream even though it's possible to set it with -ar when re-encoding it.
I filed a ticket to libav but I am just curious if there is any other way to extract sampling rate from libav probing utils. I appreciate the answer beforehand.
PS: it would be the same question for the upstream project of ffmpeg (ffprobe) in this case.