: Does a single brain lobe (e.g. parietal lobe) heat more than others during processing tasks that are related to it? Of course, our brains are the most powerful, stabilized processors. And they don't go outside of their temperature

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Short Answer- Yes, brain activity does cause temperature fluctuations in specific area.

robust changes in brain temperature could occur in a temperature-stable environment following exposure to various salient somatosensory stimuli and during different types of motivated behaviors. 
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There are different points of view on why these fluctuations occur and what their underlying mechanisms are.

Alterations in metabolic brain activity are the primary cause of intra-brain heat accumulation and a force behind more delayed changes in body temperature. 
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Brain temperature fluctuations reported in these experiments(here) correlated with the biological significance of the environmental challenges, spontaneous and stimuli-induced changes in EEG and motor activity, and had some structural specificity with respect to the modality of sensory stimuli.

Let's think about numbers:

Although the brain represents ~2% of the human body’s mass, it accounts for ~20% of the organism’s total oxygen consumption at rest 
Since all energy used for brain metabolism appears to be finally transformed into heat, intense heat production appears to be an essential feature of brain metabolic activity.

From animal experiments
All stimuli induced rapid, unique, and relatively long lasting temperature elevations in both brain structures and arterial blood, greatly exceeding the duration of stimulation.
Rat experiment

each brain structure had its own basal temperature 
stimuli induced rapid, unique, and relatively long lasting temperature elevations in both brain structures and arterial blood
temperature changes in each brain structure occurred significantly faster and had stronger amplitude than those in arterial blood
temperature changes had structural specificity. Both striatal divisions showed quite similar changes, but temperature increases in cerebellum were more delayed and prolonged following each stimulus. 
temperature changes had structural specificity. Both striatal divisions showed quite similar changes, but temperature increases in cerebellum were more delayed and prolonged following each stimulus. 

Blood supply to the brain is cooler than the brain itself:

Given that the blood supply to the brain is cooler than the brain itself, and that brain temperatures rise more quickly and to a larger extent than do arterial blood and head muscle temperatures in response to all challenges, intra-brain heat production appears to be the primary cause of functional brain hyperthermia.
While arterial blood temperature also gradually increases in response to all challenges, brain-arterial blood temperature differentials grow consistently during behavioral activation, showing an apparent increase in intra-brain heat production. Therefore, it seems that increased blood circulation that accompanies functional brain activation removes heat from the brain
brain circulation is a significant factor in the re-distribution of locally released heat within brain tissue and in its removal from the brain, thus contributing to brain temperature fluctuations occurring under behavioral conditions.
By increasing blood flow above the brain’s current demand, more oxygen and nutrients are delivered to the areas of potential demand and more potentially dangerous metabolic heat is removed from intensively working brain tissue. 

More dorsally located structures were consistently cooler

(I added this thing as I thought it was interesting here as it's opposite of what OP had in question, however, that was surface temperature)

Tested at the same conditions, more dorsally located structures were consistently cooler (dorsal striatum, 36.6°C; hippocampus, 35.6°C) than more ventrally located structures (ventral tegmental area of midbrain, 37.3°C; medial preoptic area of hypothalamus, 37.3°C; nucleus accumbens or NAcc, 37.3°C). 

In Conclusion,
in contrast to general beliefs that brain temperature is a stable, tightly regulated homeostatic parameter, data presented in this review demonstrate a relatively wide range of fluctuations (~3°C) occurring in rats following exposure to salient somato-sensory stimuli and during various motivated behaviors. Although rats and humans have obvious differences in temperature regulation and human data are very limited, similarity between rats and monkeys in general patterns of brain temperature fluctuations may support the findings.

References
1.https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3149793/#!po=22.6852