I can't see why it would be different for ectothermal animals than endothermal animals. By lower temperatures, I'm not considering a few degrees one way or the other; there is probably a maximal temperature range for the nervous system to operate, followed by decreases in lower ranges.
Some ectotherms, including several species of fish and reptiles, have been shown to make use of regional endothermy, where muscle activity causes certain parts of the body to remain at higher temperatures than the rest of the body. This allows for better locomotion and use of the senses in cold environments.
Protective mechanisms are also in place in humans, where shivering generates heat, and blood is shunted away from cold areas to core (warmer) areas to protect executive functioning (thinking) and heartbeat regularity. It doesn't mean that once thes compensatory mechanism (and others) are overcome by falling temperatures, that cognition remains unaffected.
Given extremes, there are iguanas falling frozen (not necessarily dead, though) from the trees in Florida. I doubt they even know they're falling. Maybe a muddled "Huh?" when they hit the ground, a car, etc. But I would wager that, could we read their thoughts, they are not racing with anxiety:
My ---, I'm stiff as a board! Someone help me, I can't move a muscle! What's happening? Is this a bad dream where I can't move? Am I dead? Is this HELL? Oh, ---, I wish I had spent more time with my wives and kids! Is a deathbed confession acceptable now? Can anyone hear me? Am I even talking? How could I be talking if I can't move my mouth?!
Hypothermia definitely slows down thinking in humans, e.g. in studies to try to understand the effect of cold on astronauts (not done on astronauts, though):
Chronic multifactorial stress impaired cognitive function and mood; the addition of moderate, acute cold stress further degraded vigilance and mood. When such circumstances occur, such as during disasters or military operations, measures to prevent adverse cognitive and physiological outcomes are recommended.
Cold stress is experienced in occupational (military, fishing trawlers, emergency disaster workers) and athletic (winter sports) settings (Muller et al., 2012). It appears that both moderate and extreme reductions in ambient temperature may have a negative effect on cognitive function (Banderet et al., 1986; Palinkas, 2001). Specifically, cold exposure (−20 to 10°C) has led to decrements in memory [complex task (Thomas et al., 1989; Patil et al., 1995)], vigilance [complex task (Flouris et al., 2007)], reaction time [simple task (Teichner, 1958; Ellis, 1982)], and decision making [complex task; see Table Table4;4; (Watkins et al., 2014)]. Such consistent findings across such diverse ambient temperatures may be explained by traditional theories of cold induced cognitive decrement (Teichner, 1958; Enander, 1987; Muller et al., 2012). ... Regression analysis from a recent study (Watkins et al., 2014) reported a significant relationship between alterations in thermal comfort and cognitive function in the cold.
So, yes, at sufficiently lower temperatures, cognition is decreased. One theory is that initially it is decreased because attention is directed at the discomfort of being cold (1958), but fMRI studies might clear up that issue nicely.
Cognitive function and mood during acute cold stress after extended military training and recovery.
The Impact of Different Environmental Conditions on Cognitive Function: A Focused Review