cp /etc/pulse/default.pa ~/.xsession.d/pulseaudio
chmod +x ~/.xsession.d/pulseaudio
системный сервис Вам скорее всего не нужен.

речь именно о системном сервисе

ну так unset DISPLAY, если нужно непременно из иксовой сессии
перестартовывать.

заодно расскажите, откуда у рута $DISPLAY появился.

> unset DISPLAY
так может это в скрипте делать?

> заодно расскажите, откуда у рута $DISPLAY появился.
env_keep += "DISPLAY"

$ grep -v '^\(#\|$\)' /etc/pulse/default.pa |head -1
.nofail

В любом случае, причина не в наличии $DISPLAY

покажите вывод (от рута)
pulseaudio --system --log-level 4

да нет. дело как раз в DISPLAY
после unset все нормально
# echo $DISPLAY
localhost:10.0
# service pulseaudio restart
Stopping pulseaudio service: [ DONE ]
Starting pulseaudio service: X11 connection rejected because of wrong
authentication.
main.c: read() failed: No such file or directory
main.c: daemon startup failed. [FAILED]
[root@windsor ~]# unset DISPLAY
[root@windsor ~]# service pulseaudio restart
Service pulseaudio is not running. [PASSED]                                    
                                                       
Starting pulseaudio service: [ DONE ]


(In reply to comment #7)
> покажите вывод (от рута)
> pulseaudio --system --log-level 4
> 

остановил сервис, убрал DISPLAY:
pulseaudio --system --log-level 4
main.c: Found user 'pulse' (UID 121) and group 'pulse' (GID 58).
main.c: Successfully dropped root privileges.
alsa-util.c: Using mixer control "PCM".
sink.c: created 0 "alsa_output.hw_0" with sample spec "s16le 2ch 44100Hz"
source.c: created 0 "alsa_output.hw_0.monitor" with sample spec "s16le 2ch 44100Hz"
module-alsa-sink.c: using 8 fragments of size 1408 bytes.
module.c: Loaded "module-alsa-sink" (index: #0; argument: "device=hw:0").
alsa-util.c: Using mixer control "Capture".
source.c: created 1 "alsa_input.hw_0" with sample spec "s16le 2ch 44100Hz"
module-alsa-source.c: using 12 fragments of size 1408 bytes.
module.c: Loaded "module-alsa-source" (index: #1; argument: "device=hw:0").
protocol-native.c: Allowing access to group 'audio'.
protocol-native.c: loading cookie from disk.
module.c: Loaded "module-native-protocol-unix" (index: #2; argument: "").
socket-server.c: socket(PF_INET6): Address family not supported by protocol
module.c: Loaded "module-esound-protocol-tcp" (index: #3; argument:
"auth-ip-acl=127.0.0.1;192.168.0.0/16").
socket-server.c: socket(PF_INET6): Address family not supported by protocol
protocol-native.c: Allowing access to group 'audio'.
protocol-native.c: using already loaded auth cookie.
module.c: Loaded "module-native-protocol-tcp" (index: #4; argument:
"auth-ip-acl=127.0.0.1;192.168.0.0/16").
module-zeroconf-publish.c: avahi_entry_group_add_service_strlst(): Invalid
service name
module.c: Loaded "module-zeroconf-publish" (index: #5; argument: "").
module.c: Loaded "module-volume-restore" (index: #6; argument: "").
module.c: Loaded "module-rescue-streams" (index: #7; argument: "").
x11wrap.c: XOpenDisplay() failed
module.c: Failed to load  module "module-x11-publish" (argument: ""):
initialization failed.
module-gconf.c: Unable to read or parse data from client.
module.c: Failed to load  module "module-gconf" (argument: ""): initialization
failed.
main.c: Module load failed.
main.c: Module load failed.
main.c: Daemon startup complete.

если я не ошибаюсь, это недефолтный конфиг ?
anyway, у меня с установленным DISPLAY рестартует.

приложите default.pa, попытаюсь понять, в чём дело.

Created attachment 2395 [details]
конфиг pulseaudio для работы по сети

Так же лог при ручном запуске с установленным DISPLAY
# echo $DISPLAY
localhost:10.0
# pulseaudio --system --log-level 4
main.c: Found user 'pulse' (UID 121) and group 'pulse' (GID 58).
main.c: Successfully dropped root privileges.
pid.c: stale PID file, overwriting.
alsa-util.c: Using mixer control "PCM".
sink.c: created 0 "alsa_output.hw_0" with sample spec "s16le 2ch 44100Hz"
source.c: created 0 "alsa_output.hw_0.monitor" with sample spec "s16le 2ch 44100Hz"
module-alsa-sink.c: using 8 fragments of size 1408 bytes.
module.c: Loaded "module-alsa-sink" (index: #0; argument: "device=hw:0").
alsa-util.c: Using mixer control "Capture".
source.c: created 1 "alsa_input.hw_0" with sample spec "s16le 2ch 44100Hz"
module-alsa-source.c: using 12 fragments of size 1408 bytes.
module.c: Loaded "module-alsa-source" (index: #1; argument: "device=hw:0").
protocol-native.c: Allowing access to group 'audio'.
protocol-native.c: loading cookie from disk.
module.c: Loaded "module-native-protocol-unix" (index: #2; argument: "").
socket-server.c: socket(PF_INET6): Address family not supported by protocol
module.c: Loaded "module-esound-protocol-tcp" (index: #3; argument:
"auth-ip-acl=127.0.0.1;192.168.0.0/16").
socket-server.c: socket(PF_INET6): Address family not supported by protocol
protocol-native.c: Allowing access to group 'audio'.
protocol-native.c: using already loaded auth cookie.
module.c: Loaded "module-native-protocol-tcp" (index: #4; argument:
"auth-ip-acl=127.0.0.1;192.168.0.0/16").
module-zeroconf-publish.c: avahi_entry_group_add_service_strlst(): Invalid
service name
module.c: Loaded "module-zeroconf-publish" (index: #5; argument: "").
module.c: Loaded "module-volume-restore" (index: #6; argument: "").
module.c: Loaded "module-rescue-streams" (index: #7; argument: "").
X11 connection rejected because of wrong authentication.
X connection to localhost:10.0 broken (explicit kill or server shutdown).

с вашим конфигом, с заходом через ssh -X root@otherhost -- не повторяется.

проверил еще раз на другой машине. конфиг дефолтный
вариант 1:
$ su -
# service pulseaudio restart
все нормально

вариант 2:
$ ssh -X user@localhost
$ su -
# service pulseaudio restart
ошибка.

значит, так делать не следует.