To answer this, one needs to understand a couple of somewhat independent concepts. First is that of resistance at cool temperatures.
Assuming that we are talking of a regular conductor (metal) and a temperature variation of, say, 0C to 100C, the cooler conductor will indeed have a lower resistance. Now, the question of how much lower is a separate one, but let's assume that it is measurably lower. Lets mark $R_1(T_1) < R_2(T_2)$ where $T_1<T_2$ are the temperatures.
If the same battery is connected to these resistors, we will have $I_1(R_1) > I_2(R_2)$
Now let's turn to the second concept. Namely it is that of the battery capacity. From Wikipedia:
A battery's capacity is the amount of electric charge it can deliver
at the rated voltage. The more electrode material contained in the
cell the greater its capacity. A small cell has less capacity than a
larger cell with the same chemistry, although they develop the same
open-circuit voltage.[30] Capacity is measured in units such as
amp-hour (A·h). The rated capacity of a battery is usually expressed
as the product of 20 hours multiplied by the current that a new
battery can consistently supply for 20 hours at 68 °F (20 °C), while
remaining above a specified terminal voltage per cell. For example, a
battery rated at 100 A·h can deliver 5 A over a 20-hour period at room
temperature. The fraction of the stored charge that a battery can
deliver depends on multiple factors, including battery chemistry, the
rate at which the charge is delivered (current), the required terminal
voltage, the storage period, ambient temperature and other
factors.[30]
The higher the discharge rate, the lower the capacity.[31] The
relationship between current, discharge time and capacity for a lead
acid battery is approximated (over a typical range of current values)
by Peukert's law:
${\displaystyle t={\frac {Q_{P}}{I^{k}}}}$
$Q_P$ is the capacity when discharged at a rate of 1 amp.
${\displaystyle I}$ is the current drawn from battery (A).
${\displaystyle t}$ is the amount of time (in hours) that a battery
can sustain.
${\displaystyle k}$ k is a constant around 1.3.
Using this law, you can see that for lower current, the discharge (drain) time is larger. Vice versa, for higher current, the discharge time is smaller.
In other words, $t_1(T_1) < t_2(T_2)$ or for higher temperature the discharge time is longer.
