diff --git a/compute_tools/src/monitor.rs b/compute_tools/src/monitor.rs
index 3311ee47b3..bacaf05cd5 100644
--- a/compute_tools/src/monitor.rs
+++ b/compute_tools/src/monitor.rs
@@ -13,6 +13,12 @@ use crate::metrics::{PG_CURR_DOWNTIME_MS, PG_TOTAL_DOWNTIME_MS};
 
 const MONITOR_CHECK_INTERVAL: Duration = Duration::from_millis(500);
 
+/// Struct to store runtime state of the compute monitor thread.
+/// In theory, this could be a part of `Compute`, but i)
+/// this state is expected to be accessed only by single thread,
+/// so we don't need to care about locking; ii) `Compute` is
+/// already quite big. Thus, it seems to be a good idea to keep
+/// all the activity/health monitoring parts here.
 struct ComputeMonitor {
     compute: Arc<ComputeNode>,
 
@@ -70,12 +76,36 @@ impl ComputeMonitor {
         )
     }
 
+    /// Check if compute is in some terminal or soon-to-be-terminal
+    /// state, then return `true`, signalling the caller that it
+    /// should exit gracefully. Otherwise, return `false`.
+    fn check_interrupts(&mut self) -> bool {
+        let compute_status = self.compute.get_status();
+        if matches!(
+            compute_status,
+            ComputeStatus::Terminated | ComputeStatus::TerminationPending | ComputeStatus::Failed
+        ) {
+            info!(
+                "compute is in {} status, stopping compute monitor",
+                compute_status
+            );
+            return true;
+        }
+
+        false
+    }
+
     /// Spin in a loop and figure out the last activity time in the Postgres.
-    /// Then update it in the shared state. This function never errors out.
+    /// Then update it in the shared state. This function currently never
+    /// errors out explicitly, but there is a graceful termination path.
+    /// Every time we receive an error trying to check Postgres, we use
+    /// [`ComputeMonitor::check_interrupts()`] because it could be that
+    /// compute is being terminated already, then we can exit gracefully
+    /// to not produce errors' noise in the log.
     /// NB: the only expected panic is at `Mutex` unwrap(), all other errors
     /// should be handled gracefully.
     #[instrument(skip_all)]
-    pub fn run(&mut self) {
+    pub fn run(&mut self) -> anyhow::Result<()> {
         // Suppose that `connstr` doesn't change
         let connstr = self.compute.params.connstr.clone();
         let conf = self
@@ -93,6 +123,10 @@ impl ComputeMonitor {
         info!("starting compute monitor for {}", connstr);
 
         loop {
+            if self.check_interrupts() {
+                break;
+            }
+
             match &mut client {
                 Ok(cli) => {
                     if cli.is_closed() {
@@ -100,6 +134,10 @@ impl ComputeMonitor {
                             downtime_info = self.downtime_info(),
                             "connection to Postgres is closed, trying to reconnect"
                         );
+                        if self.check_interrupts() {
+                            break;
+                        }
+
                         self.report_down();
 
                         // Connection is closed, reconnect and try again.
@@ -111,15 +149,19 @@ impl ComputeMonitor {
                                 self.compute.update_last_active(self.last_active);
                             }
                             Err(e) => {
+                                error!(
+                                    downtime_info = self.downtime_info(),
+                                    "could not check Postgres: {}", e
+                                );
+                                if self.check_interrupts() {
+                                    break;
+                                }
+
                                 // Although we have many places where we can return errors in `check()`,
                                 // normally it shouldn't happen. I.e., we will likely return error if
                                 // connection got broken, query timed out, Postgres returned invalid data, etc.
                                 // In all such cases it's suspicious, so let's report this as downtime.
                                 self.report_down();
-                                error!(
-                                    downtime_info = self.downtime_info(),
-                                    "could not check Postgres: {}", e
-                                );
 
                                 // Reconnect to Postgres just in case. During tests, I noticed
                                 // that queries in `check()` can fail with `connection closed`,
@@ -136,6 +178,10 @@ impl ComputeMonitor {
                         downtime_info = self.downtime_info(),
                         "could not connect to Postgres: {}, retrying", e
                     );
+                    if self.check_interrupts() {
+                        break;
+                    }
+
                     self.report_down();
 
                     // Establish a new connection and try again.
@@ -147,6 +193,9 @@ impl ComputeMonitor {
             self.last_checked = Utc::now();
             thread::sleep(MONITOR_CHECK_INTERVAL);
         }
+
+        // Graceful termination path
+        Ok(())
     }
 
     #[instrument(skip_all)]
@@ -429,7 +478,10 @@ pub fn launch_monitor(compute: &Arc<ComputeNode>) -> thread::JoinHandle<()> {
         .spawn(move || {
             let span = span!(Level::INFO, "compute_monitor");
             let _enter = span.enter();
-            monitor.run();
+            match monitor.run() {
+                Ok(_) => info!("compute monitor thread terminated gracefully"),
+                Err(err) => error!("compute monitor thread terminated abnormally {:?}", err),
+            }
         })
         .expect("cannot launch compute monitor thread")
 }