The default cron job daemon on Debian and Ubuntu does not support per-user timezones (see crontab(5) man page).

Here is a solution which runs hourly cron tasks in the timezone which you specified. For example, if you want to run “test.sh” at 5 AM and 7 PM in timezone Europe/Sofia, you need to create the following “cron.hourly” script:

#!/bin/bash
/path/to/run-at.sh 5,19 Europe/Sofia $(( 15*60 )) /var/run/run-at-test.state /path/to/test.sh

Here is the source code of “run-at.sh”:

#!/bin/bash
set -u

# XXX
# We work exclusively with global variables.
# Functions are used just to separate logic and for self-documenting.

function display_usage() {
	if [ "$1" -ge 5 ]; then
		return; # enough parameters
	fi

	cat >&2 <<EOF
Usage: $0 HOURS TZ WARN_TIME STATE_FILE COMMAND [ARGS...]
Runs COMMAND every day at the specified HOURS.

The execution is accounted and considered successful
only if COMMAND exits with 0. If there was no (successful)
execution within the WARN_TIME hours specified period,
a warning is being issued.

The STATE_FILE must pre-exist, so make sure that you
create it before the first run, or you will get a
warning.

This script is intended to be run in "cron.hourly".

Arguments:
 - HOURS -- comma-separated list; example: 5,12,19
 - TZ -- time zone; example: Europe/Sofia
 - WARN_TIME -- minutes; example: 360
 - STATE_FILE -- fill path to a writable file
 - COMMAND and ARGS to be executed on HOURS

Example:
 $0 5,19 Europe/Sofia \\
   \$(( 15*60 )) /root/.run-at-test.state \\
   date -d now +%c
EOF
	exit 1
}

function check_state_file_oldness() {
	# find file with mtime less than $WARN_TIME minutes
	if [ "$(find "$STATE_FILE" -type f -mmin -"$WARN_TIME")" != "$STATE_FILE" ]; then
		# file not found -> this will be indicated by 'find' on STDERR too
		# file is too old ("-mmin" condition not met)
		echo "WARNING: No successful run in the last $WARN_TIME minutes" >&2
	fi
}

function split_hours() {
	HOURS="${HOURS//,/ }" # replace "," with " "
	# https://blog.famzah.net/2013/02/17/
	#   bash-split-a-string-into-columns-by-white-space-without-invoking-sub-shells/
	HOURS=( $HOURS ) # now an ARRAY
}

function validate_tz() {
	# naive check; tested on Debian
	if [ ! -e "/usr/share/zoneinfo/$WANT_TZ" ]; then
		echo "ERROR: TZ seems to be invalid." >&2
		exit 1
	fi
}

function get_now_hour_in_tz() {
	NOW_HOUR="$(TZ="$WANT_TZ" date +%H)" # get current hour in 24h-format using the $WANT_TZ
}

function check_if_we_should_run_or_exit() {
	RUN=0
	for h in "${HOURS[@]}" ; do
		if [ "$h" -eq "$NOW_HOUR" ]; then
			RUN=1
			break
		fi
	done

	if [ "$RUN" -eq 0 ]; then
		exit 0
	fi
}

function execute_command_and_get_exit_code() {
	"$@" # execute the command
	EC="$?"
}

function update_state_file_mtime() {
	if [ "$EC" -eq 0 ]; then
		touch "$STATE_FILE"
	fi
}

#### ### ### ###

display_usage "$#"

# parse_argv

HOURS="$1" ; shift
WANT_TZ="$1" ; shift
WARN_TIME="$1" ; shift
STATE_FILE="$1" ; shift
# the rest in "$@" is the command to be executed

check_state_file_oldness

split_hours
validate_tz
get_now_hour_in_tz
check_if_we_should_run_or_exit

execute_command_and_get_exit_code "$@"
update_state_file_mtime