That means that BOINC Manager is not able to communicate with the BOINC client (which is a local (i.e. internal) connection not involving the internet).
Since no work is done the first reason which comes to mind is simply that the BOINC client is not running. So, first start the system monitor (under System->Administration) and check in the Process tab if you have a process named boinc.
If so, next we will try to find why it has suddenly stopped working.

As per Jean, open System Monitor find the boinc process of the core client (CC). There are also one or more that have a science app name associated (if the CC runs). Next start BOINC Manager (BM). Then close (not exit) the BM and observe if the CC continues. If not, open a terminal window (in Windows jargon a command or DOS window) and type

sudo /etc/init.d/boinc-client start

which starts up the CC again.

Open the BM and go to the advanced menu > options and make sure to select the 'Enable Manager exit dialog'. Toggle it off and on, then save. Assuming you use the standard Synaptic package version 6.10.17 and this is happening, you've just stumbled on an old bug. Yesterday I appended a report to an old developer trac ticket #828 that it's still happening. It's not to be fixed until client 6.12... next year I suppose.

Workaround: Just minimize the BM or first connect the BM to another host in the LAN via the Advanced menu > Select Computer, then close and the CC should continue to run.

In the old pre-service/daemon days, the BM would start the CC, but it doesn't seem to be able to do that anymore.

Question: Are you using e.g. BOINCTasks BOINCview or another BM on a different (windows) device in your LAN to monitor this Linux client?

Let us know, without feedback there not being progress to localize the source.

edit: No, BOINC has no features to shut down based on temperature. On (newer) [windows] systems you can use the TThrottle utility, to slow down BOINC when it gets too hot, in summer or when dirty inside.