pgpoolのwatchdog機能をonにし、pgpoolを起動したら、arpingコマンドの実行に失敗したというログが残りました。
[pgpool@erogamescape14 ~]$ /usr/local/pgpool-II-4.0.4/bin/pgpool -d -n > /dev/null 2>&1 &

Apr 27 22:51:46 erogamescape14 pgpool[31446]: [38-1] 2019-04-27 22:51:46: pid 31446: DEBUG:  watchdog exec interface up/down command failed
Apr 27 22:51:46 erogamescape14 pgpool[31446]: [38-2] 2019-04-27 22:51:46: pid 31446: DETAIL:  'sudo arping -U $_IP_$ -w 1' failed. exit status: 2
Apr 27 22:51:46 erogamescape14 pgpool[31446]: [39-1] 2019-04-27 22:51:46: pid 31446: LOG:  failed to acquire the delegate IP address
Apr 27 22:51:46 erogamescape14 pgpool[31446]: [39-2] 2019-04-27 22:51:46: pid 31446: DETAIL:  'if_up_cmd' failed
Apr 27 22:51:46 erogamescape14 pgpool[31446]: [40-1] 2019-04-27 22:51:46: pid 31446: WARNING:  watchdog escalation failed to acquire delegate IP
arpingコマンドを実行した結果は以下のとおりです。 
[pgpool@erogamescape14 ~]$ sudo arping -U 192.168.0.101 -w 1
Interface "eth0" is down
ErogameScapeのサーバーはeth0は使用しておらず、eth1だけを使用しています。 
[pgpool@erogamescape14 ~]$ ifconfig -a
eth0      Link encap:Ethernet  HWaddr 00:24:8C:FC:29:1E
          BROADCAST MULTICAST  MTU:1500  Metric:1
          RX packets:0 errors:0 dropped:0 overruns:0 frame:0
          TX packets:0 errors:0 dropped:0 overruns:0 carrier:0
          collisions:0 txqueuelen:1000
          RX bytes:0 (0.0 b)  TX bytes:0 (0.0 b)
          Interrupt:17

eth1      Link encap:Ethernet  HWaddr 00:1B:21:CE:8C:22
          inet addr:192.168.0.14  Bcast:192.168.0.255  Mask:255.255.255.0
          inet6 addr: fe80::21b:21ff:fece:8c22/64 Scope:Link
          UP BROADCAST RUNNING MULTICAST  MTU:1500  Metric:1
          RX packets:3514319869 errors:2 dropped:450 overruns:0 frame:2
          TX packets:3390163522 errors:0 dropped:0 overruns:0 carrier:0
          collisions:0 txqueuelen:1000
          RX bytes:1709446725964 (1.5 TiB)  TX bytes:1194077098242 (1.0 TiB)
          Interrupt:17 Memory:feae0000-feb00000

lo        Link encap:Local Loopback
          inet addr:127.0.0.1  Mask:255.0.0.0
          inet6 addr: ::1/128 Scope:Host
          UP LOOPBACK RUNNING  MTU:65536  Metric:1
          RX packets:13606420638 errors:0 dropped:0 overruns:0 frame:0
          TX packets:13606420638 errors:0 dropped:0 overruns:0 carrier:0
          collisions:0 txqueuelen:0
          RX bytes:3248906958354 (2.9 TiB)  TX bytes:3248906958354 (2.9 TiB)
pgpool.confのarping_cmdを以下のように書き換えて解決しました。
arping_cmd = 'sudo arping -U $_IP_$ -w 1'
↓
arping_cmd = 'sudo arping -U $_IP_$ -w 1 -I eth1'