Hello, I have the same problem.
Anti-Theft (rooted) is keeping my device awake all the time. It is not marked as lost. Mobile Security and Antivirus are not installed.

Do you have any ideas what prevents the Galaxy Note 3 from going to deep sleep?

Here is what BetterBatteryStats has logged (removed unnecessary info):

===================
General Information

Creation Date: 2014-03-02 08:58:25
Since 7 h 33 m 22 s
VERSION.RELEASE: 4.4.2
BRAND: samsung
DEVICE: hltexx
MANUFACTURER: samsung
MODEL: SM-N9005
OS.VERSION: 3.4.82-cm11-SM-N900x-v2.05+
BOOTLOADER: N9005XXUENA6
HARDWARE: qcom
USER: root
PRODUCT: cm_hlte
RADIO: N9005XXUENA2
Rooted: true

Battery Info

Level lost [%]: Bat.: -5% (88% to 83%) [0.7%/h]
Voltage lost [mV]: (4137-4058) [10.5%/h]

Other Usage

Screen On (): 41 s (41 s) Ratio: 0.2%
Awake (): 7 h 33 m 22 s (27202 s) Ratio: 100.0%

Wakelocks

PW com.avast.android.antitheft (com.avast.android.antitheft.Xbox One): 7 h 24 m 45 s (26685 s) Count:1 98.1%

Kernel Wakelocks !!! wakeup_sources !!!

PowerManagerService.WakeLocks (): 7 h 32 m 40 s (27160 s) Cnt:(c/wc/ec)7/0/0 99.8%
alarm (): 2 m 32 s (152 s) Cnt:(c/wc/ec)5783/0/0 0.6%
sec-battery-monitor (): 54 s (54 s) Cnt:(c/wc/ec)881/0/0 0.2%
ssp_wake_lock (): 49 s (49 s) Cnt:(c/wc/ec)2473/0/0 0.2%
qpnp-vadc-ebd45000 (): 27 s (27 s) Cnt:(c/wc/ec)5286/0/0 0.1%
PowerManagerService.Broadcasts (): 2 s (2 s) Cnt:(c/wc/ec)2/0/0 0.0%
qpnp-iadc-ebd45200 (): (0 s) Cnt:(c/wc/ec)92/0/0 0.0%
power-supply (): (0 s) Cnt:(c/wc/ec)881/0/0 0.0%
KeyEvents (): (0 s) Cnt:(c/wc/ec)1169/0/0 0.0%
flip_wake_lock (): (0 s) Cnt:(c/wc/ec)1/0/1 0.0%
event18-906 (): (0 s) Cnt:(c/wc/ec)2/0/0 0.0%
PowerManagerService.Display (): (0 s) Cnt:(c/wc/ec)1/0/0 0.0%

CPU States

300 MHz (): 7 h 33 m 99.9%
422.4 MHz (): 2 s 0.0%
652.8 MHz (): 0.0%
729.6 MHz (): 0.0%
883.2 MHz (): 0.0%
960 MHz (): 0.0%
1.19 GHz (): 7 s 0.0%
1.27 GHz (): 0.0%
1.5 GHz (): 2 s 0.0%
1.73 GHz (): 6 s 0.0%
1.96 GHz (): 0.0%
2.27 GHz (): 0.0%
2.46 GHz (): 3 s 0.0%

Many Thanks