We are in the middle of migrating from Exchange 2007 to 2013 with a single server containing both roles.
We've noticed that after about 12 hours, ActiveSync stops working for some reason. The only workaround at the moment is to recycle the MSExchangeSyncAppPool every time it breaks.

The application log is filled with the following event happening every 10 minutes or so:

Exchange ActiveSync has been loaded. This event is logged when Exchange ActiveSync receives its first sync request. The DLL that's loaded when this occurs is Microsoft.Exchange.AirSync DLL. Process id: [13604]

Is this normal behaviour? It seems as if ActiveSync is restarting constantly? This recurring event stops as soon as ActiveSync is broken. There are no other ActiveSync events in the application log.

With the help of ActiveSyncReport.ps1 I've analysed the IIS logs and for the last day it only finds about 30 people connecting and the one with the highest hits below 400. So it doesn't seem to be a faulty user device hammering the server.
I've tried the Test-ActiveSyncConnectivity command when ActiveSync is broken and this just returns the following details after 5 minutes:

Result                      : Failure
Error                       : [System.Net.WebException]: The request was aborted: The request was canceled.

The remote connectivity analyser gives the following error at the last HTTP verification step: The operation has timed out

Just to make sure, i've added 4 extra GB of ram to the server making a total of 14GB as it was running at 90% usage. But at the moment it's only using about 9GB so that doesn't seem to make a difference. Also the IIS worker processes are hardly doing anything. All together they sometimes use 3% of the 4 virtual CPU cores but mostly they stay at 0%.

Is there anything else I can try?

Are there any event logs generated when EAS stops working? This is not a normal behaviour indeed. It has to be something going wrong on the box.

I would also suggest taking a look at IIS log files at the time of failure. You should be able to see status codes at least.

I can't seem to find anything related to the stopping EAS in the event logs. While browsing through the IIS logs I could find some ActiveSync 503 errors (example below).  These error lines do seem to contain the text 'Error:StorageTransient%3a+MapiExceptionRpcServerTooBusy_Mbx:VS-Ex2013.domain.local' so maybe there is some sort of performance issue?

We are running CU2 by the way.

2013-07-26 12:36:35 fe80::cc11:3194:cbeb:66e5%12 POST /Microsoft-Server-ActiveSync/Proxy/default.eas User=barbara&DeviceId=ApplHW108QRKZ38&DeviceType=iPad&Cmd=Ping&Log=PrxFrom:fe80%3a%3acc11%3a3194%3acbeb%3a66e5%2512_V121_HH:newmail.domain.eu_SmtpAdrs:Barbara%40domain-consulting.com_Cpo18549_Fet19999_TmTr:TID%3a43%3e%3e%5bID%3aH%5FBPR%2cStart%3a12%3a36%3a15%2cEnd%3a12%3a36%3a15%2cExcl%3a0+ms%2cChild%3a%5bID%3aH%5FRMBP%2cStart%3a12%3a36%3a15%2cEnd%3a12%3a36%3a15%2cExcl%3a0+ms%2cChild%3a%5bNONE%5d%2c%5d%2c%5dTID%3a17%3e%3e%5bID%3aH%5FEPR%2cStart%3a12%3a36%3a35%2cEnd%3a12%3a36%3a35%2cExcl%3a0+ms%2cChild%3a%5bNONE%5d%2c%5d_ExStk:H4sIAAAAAAAEANVXW2%2frNgx%2bP8D%2bA5%2bGFEhd20nqpDgt0NzaAL2hSVcMw9ApspJodSRDlk%2bd%2ffpRviU%2bTZvTYN2wPjjSJ5KiyI%2bUes2pkpGcaWuQ0AURc2b1iSbWWEtFcJL%2fThQREWdCoxQLNZfiBHpECKlBhkzAkvBgKhM4kqejm%2fFoMhgfyfi0MAnn%2fpILHmlFNP%2fG4ELJOITa8Nf%2bqHt7OXQb47v%2b1eTgiIrTe0Z5aDaKzKxL1JQoYsHh4eEZXJe%2bXpOQp5%2fSnfuQjpn6xtREym4crU5gx%2fKDINOAgZbo%2bzMDKoVg1IgaSC8YRKm8BbWFOrWTtm3bTdtu1YHRU7fpdA6%2b9DmZCxlpTo22Zok%2b%2bQL4d8X9E2g67aaDExM%2fNqAnYCcdr2GWYS3UcuxWZxM4dhvuTq2m12l7FTO23Wjs3MtuNZs46QUmur8wFaVJdFqWbXmOaznehqxrt%2b3dzjsttwK4zePWLq1j2zZCa6DR8Dxv516dtt36bq%2f2Ti2n43ntTcD1Ot7Oczme17ArWq2O627X%2bslIEf0uMy9ZECKPRsgQJUgwWSj5MpoNlJLqVj0SJbiY18Za4Q8sWRRhtdUBpRsuLBSL4kDXoStlwIgAEgTyJdeJCilG67DmYi%2bjIviI9HRSh9IR4EhxVU4PPuD8ptO7fB0NkvSsM0IZ8AfxLOSL%2bLgbgwRrtldW5ZBQDP%2fK6ilGNKt9tzoSM7lRwmb63vHSTFq32LjKWXGqrOr76HAOxFFlmjc6g1zE3Ic5fq4zbBPxpxUDN2TJSsCXaERUoJBE0YtUfgkstA7vlExWWdfKhPPzDgMyLw5rxkgEc5T0GOlaVIxwpYc5iRXLArQeryn1QoS%2bZz5XaK3Y%2fmc0rwyQ7d6%2fqedtY%2bTjh%2btVZq4Cla6TMAw4JWke%2fEIREiXDbFjd%2bpFNzSacMnS2u9Lst99zw2PkV57LOkz4ko1D1Fkn2UAy1qWWZgLtXXKht6R%2b%2b%2bWWJy7fyRpKRZkJZa2kBQRcPDM%2fHWdm8SaCgfBBzjIiAyuJHWlCnwEvOSQ%2bin2mFx83OxJccxLwv9h2u3W4knOM6ipEuBilZZvavsPc4s1MAsBqZgo7DgvYHEsx1XpAjoOfI2aClJz%2biXkCXtIjK7dNn1KKwiwj6gVDs5wWdAISJzzgRK0KZN%2bDfz2jT099HoUBWfUCLDSHWF%2bzNlKVPJs%2bPTntWhUsCj6f7h%2f9rPtv27f2TwXezyL%2fZsnnBZrXLUIZgC27G%2ftzpmGa%2ftThPC2wPxxkeEGaqsum6W9fGeIMN3xTM1%2bvQxUeL%2fCZh%2b8xE8NrIjCCCiLEmL%2bB7Bv8Tw36fmz%2feI7wNsCbWK%2f6bJbGVgrseTk58zVueujoCt%2fZmLqZDHx85E2kydQPlNdrFvybGZJixudxukHa%2bl4nYllcsrnqOcUrI0oDSMrhPoHdSoQfjnZB8%2fS%2by1JsFPLb941k%2fIeRNtF95HrRZZHOQvh2rG8z7hf3eiqNo4f8RdTLr2kaK%2bRP8e6k29B98vJpF8L%2fJQTvne%2bcq%2fFKUCyc5ZII33rdbmq5SNqszBv0I%2bYepXrG%2f5XxTU%2f82sHfOyI1oCQQAAA%3d_Error:StorageTransient%3a+MapiExceptionRpcServerTooBusy_Mbx:VS-Ex2013.domain.local_Throttle1396_TmRcv12:36:15.9800986_TmSt12:36:17.3841327_TmFin12:36:17.4309311_TmCmpl12:36:35.9797709_NmDeferred1_ActivityContextData:I32%3aADS.C%5bVS-Legolas%5d%3d2%3bF%3aADS.AL%5bVS-Legolas%5d%3d6.6164%3bDbl%3aBudgUse.T%5b%5d%3d46.7984008789063%3bS%3aWLM.Cl%3dCustomerExpectation%3bS%3aWLM.Type%3dEas%3bS%3aWLM.Int%3dTrue%3bS%3aWLM.SvcA%3dFalse%3bS%3aWLM.Bal%3d232154%2c6%3bS%3aWLM.BT%3dEas_Budget:(D)Id%3adomain%5cbarbara%5FApplHW108QRKZ38%5FiPad%2c100%25%2cLastUpdate%3a26%2f07%2f2013+12%3a36%3a15%2cMaxConn%3a10%2cConn%3a0%2cMaxBurst%3a240000%2cBal%3a240000%2cCutoff%3a600000%2cInterCallsFiveMin%3a0%2cCallsFiveMin%3a19%2cLiveTime%3a00%3a00%3a20.4208897%2cActiveDevices%3a1%2cThrottlingPolicy%3aGlobalThrottlingPolicy%5Fb61dcd62-4cf8-485e-8685-344c2af76069_ 444 domain\barbara fe80::cc11:3194:cbeb:66e5%12 Apple-iPad1C1/803.148 - 503 0 0 33072

Looks like the client is throttled. I would recommend checking your mailbox server performance. You said, you have to recycle the app pool and it starts working. I presume, the pool does not crash. If that is true, it is nothing to do with active sync as a service though.