If a Tstat stucks open, I would replace it promptly. You're not only waste fuel but also will not have heat. The engine will never get to normal operating temp which is bad for internal part.
To check for vacuum leak, bring it to a shop and do a smoke machine test. The plastic intake tube crack that cause vacuum leak is very common.

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00 E39 DINAN S1 //M5 (82k miles)
06 E53 DINAN 4.8iS, born 2/18/2006 (126k miles) http://www.xoutpost.com/bmw-sav-foru...s-my-dslr.html
90 Straman Z32 TwinTurbo Convertible, (1 of 44 ever made) 5 speed, 444rwhp/451rwtq
01 360 Novitec Spider F1 (26k miles)