How does the brain regulate its temperature?
From what I can make up from your question is that you assume the blood in the brain is a bad thing in terms of temperature control. It is not. On the contrary, blood circulation is crucial in maintaining brain temperature.

Under conditions where the brain is heating up, which seems the focus of your question, there are mechanisms in place to extract this heat and dissipate it. Ambient temperature is often lower than body temperature, and even in hot climates, transpiration and water evaporation may cool the skin nonetheless.  

Indeed, during hyperthermia, venous blood flow from the skin of the face and scalp fed to the dura mater (the brain's membranes) is enhanced and cools the brain (Brinnel et al., 1989). Moreover, blood from the nasal and paranasal mucous membranes flowing to the dura mater cools the brain due to the evaporative cooling of sweat or mucus. The dura mater may also transmit temperature changes to the cerebrospinal fluid compartment (Zenker &amp; Kubik, 1996).

References
- Brinnel et al., Arch Dermatol Res (1989); 281(1): 66-72
-  Zenker &amp; Kubik, Anatomy Embryol (1996); 193(1): 1-13